bela bollobás, oliver riordan-percolation -cambridge university press (2006)

Percolation

Percolation them y was initiated some 50 years ago as a umthernatical ft aureworkfor the study of random physical processes such as flow through a disorderedporous medium. It has proved to be a remarkably rich theory, with applicationsbeyond natural phenomena to topics such as the theor y of networksMathematically, it has many deep and difficult theorems, with a host of openproblems remaining The aims of this hook are twofold. Fitst to present classicalresults. including the fundamental theorems of Harris. liesten, IVIerishikra,Aizenruan and Newman. in a way that is accessible to non-specialists. Theseresults are presented with relatively simple proofs, making use of combinatorialtechniques Second. the authors describe, fox the first time in a book recentresults of Stith/70V on conformal invariance, and outline the proof that thecritical probability for Voronoi percolation in the plane is 1/2

Throughout. the presentation is streantlined, with elegant andstraightforward 'nook requiring minimal background in probability andgraph theory. so that readers can quickly get up to speed. Numerousexamples illustrate the important concepts and enrich the arguments All inall. the book will be an essential purchase lot mathematicians probabilists,physicists, electrical engineers and computer scientists alike

y

Percolation

BELA BOLLOBASUniversity of Cambridge and

UM versify of Memphis

OLIVER RIORDANUni vet sit y of Comb; idge

IL.6(= CAMBRIDGE.‘cuy UNIVERSITY PRESS

caauuunora UNIVERSII i PRESS

Candwidge New York . Melbourne . Nlathid. Cape Town. Singapore,Silo Paulo

Cambridge University PressThe Edinburgh Building . Cambridge C132 2B U.. UK

Published in t he United States of America by Cam bridge University Press . New York

www cambridge orgInformation on this title: www cambridge mg/978052187232,1

Cambridge University Press 2006

this publication is in copyright Subject to statutory exceptionand to the pro \ owns of relevant collective licensing agreements

no reproduction of any part may take place withoutt he written permission of Cambridge University Press.

First published 2006

Printed in the United Kingdom at the University Press Cambridge

A catalogue teToui for this publication is available ban: the ilish Lanaia

.1SBN-13 078-0-521-87232-4 11 n 11(11MCI:

ISBN-I0 0-521-87232-4 hardback

Cambridge University Press has no responsibility flu the persistence or accuracyof URts for external or third-party intemnet websites referred to in this publication,mid does not guarantee that:my content oil such websites is or will remain, accurate

or appropt hate

To Gabriella and Gesine

Contents

Preface page ix

1 Basic concepts and results 1

2 Probabilistic tools 36

3 Bond percolation on V — the Harris-Kesten Theorem 50

:133.1 The Russo--Sem i an-Welsli method

53 2 Hai 1 is is Them mu 63.3 A sharp transition

(

3.4 Kesten's Theorem 673.5 Dependent percolation and exponential decay 70:3 6 Sulr-exponential decay 76

4 Exponential decay and critical probabilities - theo-rems of Menshikov and Aizenman Sz Barsky 784 1 The van den Berg-Kesten inequality and per colation 784 9 Or iented site percolation 804 3 Almost exponential decay of the adius Menshilaw's

Theorem 904.4 Exponential decay of the radius 104

Exponential decay of the volume the Aizenman--Newnmn Theorem 107

5 Uniqueness of the infinite open cluster and criticalprobabilities 1175 1 Uniqueness of the infinite open cluster - the

Aizenman-Nesten-Newman Theorem 1175. 9The Mann is-Kesten Theorem revisite d ! 1245 3 Site per colation on the triangular and square lattices 1295.4 Bond percolation on a lattice and its dual 1:36

vii

viii Contents

5.5 The star-delta transformation 148

6 Estimating critical probabilities 156

6.1 The substitution method 156

6.2 Comparison with dependent percolation 1626.3 Oriented percolation on Z 2 1676..1 Non-rigorous bounds 175

Conformal invariance — Smirnov's Theorem 1787 1 Crossing probabilities and conformal invariance 178

72 Smirnov's Them em 1877 3 Critical exponents and Sch in n-Loewner evolution 232

8 Continuum percolation 240

8 1 The Gilbert disc model 2418.9Finite random geometric graphs 254

8 3 Bandon' Voronoi percolation 263Bibliography 299Index 319List of notation 322

Preface

Percolation theory was founded by Broadbent. and Hammer sley [1957]almost half a century ago; by now, thousands of papers and many hookshave been devoted to the subject The original aim was to open up tomathematical analysis the study of random physical processes such asthe flow of a fluid through a disordered pm MIS medium These bone

fide problems in applied mathematics have attracted the attention ofmany physicists as well as pure mathematicians, and have led to theaccumulation of much experimental and heuristic evidence for manyremarkable phenomena. Mathematically, the subject has turned out tobe much more difficult than might have been expected, with several deepresults proved and many more conjectured

The first spectacular mathernatical result in percolation theory wasmoved by ICesten: in 1980 he complemented Harris's 1960 lower boundon the critical probability for bond percolation on the square lattice, andso proved that this critical pr obability is 1/2 To present this result, andnumerous related results. Kesten [1982] published the first monographdevoted to the rnathematical theory of percolation, concentrating onchscrete two-dimensional percolation A little later, Chayes and Chayes[1986b] came close to publishing the next book on the topic when theywrote an elegant and very long review article on percolation theory un-derstood in a much broader sense

For nearly two decades, Grimmett's 1989 book (with a second edi-tion published in 1999) has been the standard refer ence for !larch of thebasic theor y of percolation on lattices Other notable books on variousaspects of percolation theory have been published by Smythe and 'Wier-man [1974 Durrett [19881, Hughes [1995; 1996] and Meester and Boy[19961; valuable survey articles have been \\ ridden by Durrett [1984],

Preface

Chaves. Puha and Sweet (1990]. I:esten [200:31 tun! Gti uu uett 12001]among others.

Out aims in this book me two-fold 11 hst. we aim to present theresults of percolation in a way that is accessible to the non-

specialist To get straight to the point. we shut with the best knowntesult in the subject, the fundamental theme/It of limits and ICesten.even though this is a special case of late' and mule genet al results givenin subsequent chaplets

The moot of the Han is Kesten Theorem. in pal t iculat the tippethound clue to kesten. was a peat achievement, and his pool not simpleSince them however, especiall y with the advent of new tools in proba-bilistic combinatmi CS aunty Sininle moors hate been B er ard.. a fact thatmost non-specialists ale not awn' e of Pot some of these at gutnents. allthe pieces lute been published some time ago, but pet haps not in oneplace. in only as comments that ale easy to ndss Here. we In ing to-Reilly' these ions pieces and also mole tecent. ver y simple proofs ofthe Han Nesten Theorem

In Chapters -1 and 5 we desct the t he vety genet trite:tuns of Nlenshikov,and of Aizemnan kesten and Newmn: these Jesuits ale again classicalOin aim hew is to present them in the greatest genetalit \ that does notcomplicate the proofs

Out second aim is to present recent results that have not let appearedin book fie.i t s Nit give a complete moo( of Sinn no\ 's famous confonnalinva t bu t te result: to out knowledge no such account. tvitlt Hie Is clot-ted a nal t fl ossed. has previously appeared We finish by pwsentingan outline of the recent moot that the critical probabilit y for randomVotonoi percolation in the plane is 1/2

As is often the case. we have flied to mite the kind of book we shouldlike to wad I3' now percolation theory is an immensel y rich subjectwith enough material lot a dozen books. so it is not smptisiug thatthe choice of topics sttonglv reflects out tastes and intetests We havestriven to give stleatulined moots and to Ming oat the elegance of thearguments. To make the book accessible to as wide a ! cadetship aspossible. we have assumed NT/ 1 littlelittle mathematical background and haveillustrated the impol taut concepts and main arguments with mune/ ousextunples

In niit ing this book we lime receiNed help fron t ma i n people PaulBalistet. Gesine Ciosche Henn Liu. Robert tklol r is. Anthem Sat kat andNIB/ k lkaltels were kind enough to tend parrs of t he manuscr ipt and tocorrect many misprints: lot the many that iemain. we apologize

1

Basic concepts and results

Percolation thecny was founded by Broadbent and Flantnansley f19571.in cruder to model the flow of fluid in a porous mediurn with randomlyblocked channels. Interpreted narrowly, it is the study of the componentstructure of random subgraphs of graphs Usually, the underlying graphis a lattice or a lattice-like graph, which may or may not be oriented,and to obtain our landon, subgraph we select vertices or edges indepen-dently with the same probability p Iu the quintessential examples, theunderlying graph is Ed

The aim of this chapter is to introduce rile basic concepts of percola-tion theory, and some easy fundamental results concerning them.

\\k_r shall use the definitions and notation of graph theory' in a stan-dard way, as in Bollobtis [19981. for example In particular, if A is agraph, then V(A) and E.:(A) denote the sets of vertices and edges of A.respectively "re write E A for :r E V(A) We also use standard nota-tion for the limiting behaviour of functions: for / = /(n) and g = g(o).

we write = o(g) if //q as = 0(g) it fl g is bounded,= n ( g ) for g = 0(f), and f = e(g) if / = 0(g) and q= 0(.1')The standard terminology of percolation theory differs horn that of

graph thew y: vertices and edges ale called sites and bonds, mid com-ponents are called clusters When our random subgraph is obtained byselecting vertices, we speak of site percolation; when we select edges,bond pen:olation In either case, the sites w bonds selected are calledopen and those not selected are called closed; the state of a site or bondis open if it is selected, and closed otherwise (In some of the earlypapers, the term 'atom' is used instead of 'site', and 'dammed' mid 'un-dammed' for 'closed' and 'open' ) In site percolation, the open sabgroph

is the subgraph induced by the open sites: in bond percolation, the open

subgraph is Mimed by the open edges and all vertices; see Figure 1

o •

• •

o o • 0 •

o

o

o

•

0

•

o 0 0

0

I I I

2 Basic concepts and results

Figure 1 Parts of the open subgraphs in site percolation (left.) and bondpercolation (right) ) on the square lattice 1 2 On the left, the filled circles arethe Open sites; the open subgraph is the subgraph of 2: 2 induced by these Forbond percolation. the open subgraph is the spanning subgraph containing allthe open bonds

To streamline the discussion, we shall concentrate on mu)/ iented per-

colation, i e on (bond and site) percolation on an unoriented graph A

\Nre assume that A is connected, infinite, and locally finite (i.e., every

vertex has finite degree) In general, A is a in ulti-gurph, so multiple edges

between the same pair of vertices me allowed, but not loops Most of

the interesting examples will be simple graphs

Often, we shall choose bonds or sites to be open with the SUMO proba-bility p, independently of each other This gives us a probability measureon the set of subgraphs of A; in bond percolation we write Pen for thismeasure, mid in site percolation P sA p More often than not, we shallsuppress the dependence of these measures on some or all parameters,and write simph IF or Pp Similarly, A ll ; is the open subgraph in bondpercolation, and in site percolation

Formally, given a. graph A with edge-set E, a (bond) configuration isa function : E {0, c we ; we write Q = {0,1} E lot the set ofall (bond) configurations A bond e is open in the configuration w if andonly if = 1, so configurations correspond to open subgraphs Let E

be the a-field on Q generated by the cylindrical sets

C(P, a) = {co E Q f = a f lot 17} ,

where F is a finite subset of E and a E 10, I F Let p = with•0 < pc < 1 fin every bond c We denote by Pn p the probability measure

I Basic concepts and results

on (12 E) induced by

P,1,(C(F,e7)) PI fl (— p f)

(1)IEr fEbaf =1 af=0

When pc = p for every edge e, as before, we write P,I4 for PbAp

In the measure P!‘) ,p , the states of the bonds are independent, withthe probability that e is open equal to p„; thus, lot two disjoint sets Ehand F 1 of bonds,

Pi Ithe bonds re open and those in Fa are closed)

= Pf 11fe ro

We call I1114 an independent bond percolation IIICatilln on A The special

case where P. = p for every bond c is exactly the measure P IA' ,1, clefinedinformally above. The formal definitions for independent site percolationare similar

Let us remark that site percolation is more general, in the sense thatbond percolation on a graph A is equivalent to site percolation on L(A),the tine graph of A This is the graph whose vertices are the edges of A;two ve t tices of L (A) are adjacent if the corresponding edges of A sharea vet tex: see Figure 2

Although in this chapter we shall make some remarks about generalinfinite graphs, the Main applications are always to lattice-like' graphsThese graphs have a finite number of 'types' of vertices and of edgesOccasionally, we may select vertices or edges of different types withdifferent probabilities

For a fixed underlying graph A, there is a natural coupling of themeasures il-^1 .1„ 0 < p < 1: take independent random variables „Y„ foreach bond c of A, with X„ uniformly distributed on [0, We mayrealize Ai; as the spanning subgraph of A containing all bonds e withX„ < p. In this coupling, if p i < p), then is a subgraph of A. Asimilar coupling is possible for site percolation

An open path is a path in the open subgraph For sites 17 and if wewrite 'II or {x —4 yl for the event that there is an open path hornx to y, and P(x y) for the probability of this event in the measureunder consideration We also write ',/r for the event that there isan infinite open path starting at x.

An open cluster is a component of the open subgraph As the graphs

3usic Concepts and rcvalts

Figure 2 Part of t he squaw lattice L 2 (solid circles and lines) and its linegraph /3(2) (hollow circles and dotted lines) Note that ii,( 2: 2 ) is isomorphicto the non-planar graph obtained from E 2 by adding both diagonals to everyother face

we consider are locall y finite.. an open cluster is infinite if and only if,for every sites in tile cluster the et-cut holds. Given a sitewe r\ rite Cr for the open cluster containing ;r.. if there is one; otherwise.we take Cs to be erupt) Ilms Crix = {q E A: a! ---rr y} is the set of sitesq for which thew is an open Path Clearly, in bond percolation, exalways contains x, and in site percolation. = N if and 011/1' if ,rt isclosed.

Let Ox (n) beet he p t obahilih that C.,. is infinite. so 0, r (p) = Pvt./7Needless to sax, Ox (p) depends on the underlying graph A. and whetherwe take bond or site percolation More finmalty, Eat bond percolation,for example,

= 0, , (A = 0,1; (A: p) =11') 1 ,(1Cr i = c.c.)

where 11 ' ter is the ntunbet of sites in Cr We shall usewhichever !bun of the notation is clearest in an y given context In future.we shall introduce such self-explanatory \ rat rants of our notation withoutblither comment: we believe that t Iris will not lead to confusion Twosites r and y of a graph A rue equivalent if thew is an automolphisniof A mapping .r to a \ hen all sites ate equivalent (i.e. the svmmet tS

.1 Basic concepts mid orsalt 5 5

group of the graph A acts transitively on the vet tires). we write 0(p) for

Or (p) fa any site x The quantity 0(p), or Ox (p), is sometimes known as

the percolation probability

Clearly, if a, and y are sites at distance il, then (E(p) > p''U,(p), so

either Orr (p) = 0 for every site x, or 0,.(p) > 0 for every x Trivially.

from the coupling described above. 0,.(p) is an increasing function of pThus there is a critical probability pn, 0 < < 1. such that if p < prr,

then ()Apr = 0 For every site and if p > pH , then (E(p) > 0 for

every ,r The notation p H is in honour of thimmersle) \Viten the modeltinder consideration is not clear horn the context, we write pil l (A) for

site percolation on A and 14). 1 (A) for bond percolationThe component str ucture of the open subgrirph undergoes a dramatic

change as p increases past pH : if p < pi] then the probability of theevent E that there is an infinite open duster is 0, while for p > pp thisprobability is 1 To see this, note that the event E is independent of thestates of /WV finite set of bonds or sites. so Kohnogorey 's 0-I law (seeTheolem 1 in Chapter 2) implies that Pp (E) is either 0 or I. If p < pH,

So that 0 ( p) = 0 for Oyer X, then

filv (E) 0,(p)-= 0

""d P > Th r • the" Py( E) > OAP) > 0 lo t some site (and so forall sites), implying that. Pp (E) = I. One SUNS that percolation

in a certain model if 0,(p)> 0. so P11 (E) = t \kith a slight abuse ofterminoloro, we use the SlUtle word both for this particulat event andfor the measures studied; this is not ideal. but. as in so man) subjects.the historical terminolog y is now entrenched

To start with, the theory of percolation was concentric) mostly Willi tile

Stull of critical probabilities, i e with the question of when percolationoccurs NOW, howevet, it encompasses tire study of much mole detailedproper ties of the random graphs arising fron t percolation measures Infact, great efforts are made to describe the structure of these randomgraphs at or near the critical probability. eve" Wile/1 we cannot pin downthe critical meltability itself In Chapter. 7, we shall get a glimpse of thehuge amount of work done in this area, although in a setting in whichthe critical probability is known

The theoo of percolation deals with infinite graphs.. and 110111V ofthe basic events stitched (such as the occurrence of percolation) in-volve the states of infinitel y UtiUle bonds Never tireless, it always suf-[ices to consider events in finite probability spaces, since, for example.

b r (p) = Pp(IC,) > a). In this book, almost all the time., eventhe definition of the infinite product measure will be irrelevant

Fm p < p H , the open cluster Cr is finite with probability 1, butits expected size need not lie finite This leads us to another criticalprobability, p i , named in honour of Temper ley Again, we write K(A)or p1/4 (A) for site or bond percolation on A. For a site rr, set

NAP) = Es(ICA),

where E t, is the expectation associated to IF,, If all sites are equivalent,we write simply ixr(p). Trivially, x x (p) is increasing with p, and, as before,vr (p) is finite for some site rr if and only if it is finite for all sites. Hencetheme is a critical probability

Pr = suP{p: iVr(P) < oc } = (p) = cob

which does not depend on X. By definition, p i. < One of our aimswill be to prove that pr = pp for many of the most interesting groundgraphs, including the lattices Zd , d > 2

There rue very few cases in which p it and pt are easy to calculateThe prime example is the d-regulal infinite fate, otherwise known as the.Bethe lattice (see Figure 3). For the pm poses of calculation, it is more

Figure 3 The 3-regular nee, for which A = .1 ), = A = 1ii i = 1/2 Deletingan edge (e, for example), this tree falls into two components, each of which isa 2-brunching tree

convenient to consider the k-Inanching treeTk This is the rooted tree inwhich each vertex has k children, so all sites but one have degree k 1Wm Ring am for the root of I. let „ be the section of this tree up to

I Basic concepts and results 7

Figure 1 The tree J . 1 w i th root vo

height (or, following the common mathematical convention of plantingtrees with the root at the top, depth) o, as in Figure Taking thebonds to be open independently with probability p, let 7" = (p) hethe probability that Tk , „ contains an open path of length o from theroot to a leaf Since such a path exists if and only if, for some child in

of vo, the bond nom is open and there is an open path of length n — 1

horn v i to a leaf, we have

IT„ = - (1 - int„ i) k = jk pkn-I )

(2)

On the inter val [0,4 t he function Tr; p (x) is increasing and concave, withfk,i ,(0) = 0 and fk, ,,(1) < 1, so h.. 1 ,(a 0 ) = iro for some 0 < :co < 1 if andonly if r (0) = kp> 1; fur thermore, the fixed point yo is tmique whenit exists (see Figure 5) Thus, if p > 1/k, then, appealing to (2) we see

Figure 5 For k = 2 and p = 2/3, the increasing concave function fix) == — ill' satisfies = 4 1(0) = 0, J(3/1) = 3/1 and

1(1) = 8/9

S Basic concepts and results

Thal )r„_ > up implies tr„ > .to Since rut = I, it follows that rr„ > .rp•

fbi ewes' so 0;1 (,)/ k : > ./ . 0 > 0. implying that ph (Tr) < 1/k Also ifp < 1/k. then 7„ ()olive/gem to 0 11w unique fixed point of 1 k,i,(3). andso 0/),),,(Thip) Hence. the critical probability p li 'I (Ik ) is equal to 1/k

Tinning to p i note that the probability that a site p at graph distance

I from the toot up belongs to is exactl y ji Thus

\ I:.„(11 E(1C/01 E y; opt.„Erk ,=0

which is finite tot p < 1/k and infinite lot p 1/k Thus the criticalmobabiliB p!; (TA ) is also equal to 1/k

Fin any infinite Bite. fillet conditioning on the toot .1 being open. the

open clusters containing r in site and bond percolation have exactly the

same distribution. Indeed, each child of a site in the open cluster lies

in the open cluster with probability p for the k-branching BeeT), we have p7 i = = = lrlr = 1/k It is easy to show similatly

of indeed to deduce. that the loin critical probabilities associated to the

(A) -I-I )))1 egulat nee ate also equal to 1//,

The a t gu i nea' about innounts to a comparison between percolation on

Tr, and a «ti lain branching process; we shall give a slighth less trivial

example of such a compaiison shot tl y If A is il/IN graph with maximumdegree then a one-wit\ comparison wit blanching !actress showsthat all ethical l a °liabilities associated to A a l e at least I,/(A – 1) Tosee this mo t e easil y. note that tot every p E C, , tin g e is at least one openpath in A Irotu i` to p Thus ) (p) = is at most the expected

number of open (finite) paths in A starting at .r Them are at most

A(A – 1 paths in A of length I slatting at ,r. so

V,'( p ) + – 1)//5/1st

and

1)(-1p1+1

fin bond and site petcohttion espect ivelv Both sums converge lo t anyp < l/(L\ – Pit)(A) /4 ( A ) > – As PH > t i . the cone -sponding inequalities loi m i follow This shows that among all giaphs

with maximum degree the A-1(4mila' tree has the lowest et itical prob-abilities

Thew all' VW lolls tit ivial chan ges we can mid«) to a gi aph whose effect

1 Basic cancel& curd results 9

on the critical probability is easy to calculate. For example, if A is

any graph and A ( ' is obtained from A by subdividing each edge I. - 1

times, then 14::(A(0 ) = where pl.' is p lA or Also, ifis obtained from A by replacing each edge by k parallel edges. then

= (1 -pi,1(A))Uk, where p eli is ph or Of comse, pst,(A ild ) =pas (A). Combining these operations. we may replace each bond of a graphby k independent paths of length (..• to obtain a new graph For bondpercolation, the critical probabilities po w and p„„„. satisfy

1 - (1 - mcm „. = p„rd

In this way, by a trivial operation on the graph, a critical probability in

the interval (0,1) can be moved very close to any point of (0,1)

If we know the critical probability for a graph A, then we knor, in-stantly the critical probabilities Mr a family of graphs A' obtained bysequences of trivial operations flour A, as in Figure 6

Pignut ti tornsfor wing one bond petcolation model into another, and their

into a site percolation model If the first (the hexagonal lattice) has criticalprobability p, then the second has critical probability I sat isfying 1 3 (2— r ) = p

which is also the critical probability For site percolation on the third graph

It is easy to show that any 0 < < 1 is the critical probabilit y forsome graph, indeed, for some tree Let T be a finite rooted tree withheight (depth) h, with (" leaves. Let T I = T. and let T" be the rootedtree of height ha formed from T"- 1 by identifying each leaf with the

root of a copy of For example, if T is a star with k edges, then T"

is the tree Tk ,, defined above Let 1' be the 'limit' of the tacos T".defined in the obvious way.

Taking the bonds of 7' to be open independently with probabilit y p.the nunibm of leaves of T joined to the toot by open paths has a certaindistribution X with expectation ph ( Now suppose that the bonds of T'are open independently with probability P. and let X„ be the number

LU Basic concepts and results

of sites of Tux at; distance fern horn the root joined to the root by openpaths Then the sequence (X0 , Xr, .) is a branching process: wehave X0 1, and each X„ is the sum of X„_ i independent copies of thedistribution X As X is bounded, excluding the trivial case p = C = 1,it is easy to show (arguing as above flu Tk ) that percolation occurs ifand only if E(X) > 1, i e., if and only if p fi e > 1: this is a special caseof the fundamental result of the theory of branching processes In fact,one obtains

14. (I') ) = 1, 11.'(Tcc ) = (3)

Suppose now that k > 1 and 1/(k 1) < 7 < 1/k Define (1 < a < 1by (k-r-1) a k 1 -0 = 1/7 Let a = La i be the 0-1 sequence with densitya constructed as follows: whenever divides i but 2-) does not. set

= 1 if and only if the )th hit in the binary expansion of a is 1. Let'ZO be the tooted tree in which each site at distance i lion/ the root hask+arr. i children R. is easy to check that, for each n, we can find trees 77!and T" of height C = 2" such that (21's c C (T")" , where T" has(k ± 1)/k times as Mai/ \ leaves as Using (3), one can easily deducethat p„(Ta ) = 7, where p c denotes any of the four critical probabilitieswe have defined

Alternatively, let 7 be the t andom tooted tree in which each site hask + 1 children with probability r and /i r children with probability 1 - r,with the choices made independently for each site It is easy to showthat with probability 1 this random tree has p i .(T) = +1).

In general, it is easy to calculate the various critical probabilities for agraph that is 'sufficiently tree-like'. For example, for C > > 3, let Ckj

be the cactus shown it/ Figure 'i This graph is formed by replacing eachvortex of the k-regidar tree Tr; by a complete map(' on C vertices, andjoining each pair of complete graphs col esponding to an edge of TA. byicier/allying a vertex of one with a vertex of the other, using no vertex inmore than one identification, We call the vertices resulting from theseidentifications attachtrient vet Nees Although Gek,t contains many cycles,it still has the global structure of a tree, and percolation on CiL i mayagain be compiled with a blanching process

Indeed, let K; be a complete graph with k distinguished (attachment)vertices v l . Taking the edges of lid to be open independentlywith probability p. let .Vi, be the random number of vertices among

. irk that may be reached hum lir by open paths Let us explorethe open cluster of a given initial site .r of Cr c by working outwardsfrom :r Except at t he first step, how each attachment vertex that we

. Basic concepts and results 11

Figure 7 Part of the cactus C3 r: each circle represents a complete graph onvertices Where two circles touch, the corresponding complete graphs shale

an 'attachment' vertex.

reach at; a given step, the number of further attachment vertices that.we reach at. the next step has the same distribution as X i„ and thesenumbers are independentt. It follows, by considering a branching processas above, that

P i'l ( C5c,O = ( Ck = lid (1) E(X11) >

This quantity may be easily calculated for given k and The criticalprobabilities for site percolation on Ckj may be calculated in a corre-sponding way Furthermore, there is nothing special about completegraphs: there me many similar constructions of 'tree-like' graphs forwhich the critical probabilities can he calculated using a branching pro-cess.

In the rest of this chapter, we shall prove some easy bounds on thevarious critical probabilities for 7L", d > 2 Of course. percolation on Z

is trivial: all the critical probabilities we have defined are equal to 1In studying 72 , we shall make use of simple properties of plane graphs

As all such graphs we consider will be piecewise-linear (in fact, they willhave subdivisions that are subgraphs of 2Z 2 , if we wish), there are notopological difficulties In fact, all we shall need is that every polygon(in the graph Z-, say) separates the plane into two components, the

Basic concepts and results

Thief in; and the el:lei-On, with the interior bounded (This is easily seenby considering the winding number of the polygon. which changes 1w1 when we cross a side ) F l our this, Euler's [bra nch follows easily byinduction (see Chapter 1 of Boflobtis (19981 for the details), which innun implies that IS:5 and A.3 3 are non-planar Thus, for example, if Cis a cycle in the plane and a, b. c, d are four vertices of C appearing inthis order around C', then neither the interior nor the exterior of C cancontain disjoint a- c and b-d paths; see Figure 8

Figure 8 Neither t he interim tun t he exterior of a cycle visiting a, b c d int his older Can contain disjoint a--c and b d paths (solid lines) Both statementscan be deduced either fron t the non-planarity of h, or hont t hat of ICA a byconsideting t he clashed lines

Let us point out an important : sell-duality property of This willbe very important later: now we shall use it in a trivial way The dualA t of a graph A drawn in the plane has a vertex for each face of A,and an edge e' fin each edge c of A: this edge joins the two ver ticesof A' corresponding to the faces of A in whose boundm y e lies WhenA = T2 , it is custormuy to take = (3, ± 1/2, y±1/2) as the dual -Vet texcorresponding to the face with vertices (x.y). (:r+ Ty), (:r+ 1, y -k I),

y+ 1): see Figure 9 dual lattice is then E2 (1/2, 1/2), whichis iti011101phic to A. The dual graph A* is important in the context ofbond percolation on A In this context, the sites and bonds of A' areknown as dual sites and duel bonds. and a dual bond e ." is usually takento he open when c is closed and vice versa

One trivial use of planar duality is to prolific an alternative way tovisualize site percolation Let A be a plane graph: in face percolation

on A. we assign a state. open 01 closed, to each face of A The facesof A form the vertex set of a graph in which two faces are adjacent ifthe ) share an edge: this graph is precisely A.*: see Figure 10 Thus. face

0-

0-

0-

.1 Basic cconcept—,s and re5ults 13

tMIENII=MEM t

° MINFigure 9 Portions el t lie latti ce A (solid law s ) awl the isomorphic duallattice A • (dashed lines)

Figure ID lace percolation on the hexagonal lattice (left): the open facesare shaded. The corresponding open subgraph in the site percolation on thetriangular lattice is shown on the right (to the saute scale!).

percolation on A is equivalent: to site percolation on A' Thinking ofopen faces o 1 sites as colotned black, fin example, and closed faces orsites as white, the face percolation picture is easier to visualize

Returning to the study of percolation on Z2 , if H is a finite connected

submaph of A = Z2 with vertex set C. then there is a unique infinitecomponent C of A — the submaph of A induced by the vertices

outside H. By the external boanda i y C of C we mean the set of bonds

of A* dual to bonds of A joining C and Cc.; see Figure 11. Out firsttask is to show that D"C has the pope/ties we expect of a boundary

14

Basic con its and results

0 0 0 0 0 0 0 0 0 0 0

Figure 11 A finite connected subgiaph H of Z2 with vertex set C (solid linesandcircles). Vertices in finite components of 2 2 — C are shown with crosses.(some of ) those in the infinite component with hollow circles- The dotted linesme the C--C, bonds, and the dashed lines the external boundary of C

Lemma 1. If C is the vertex set of a finite connected subgraph of V,

then 0"C i5 a cycle with C" rn ils inle t iot

Proof._ Let _F be the set of C-C, bonds, oriented flow C to Ct. Emf = ab E F, orient the dual bond so that a is on its left, to obtain an

oriented dual bond f = Equivalently, f is f caged counter-, —clockwise through r/2 about: its midpoint Let a-0={ 7 E F},so O'C' is an orientation of O'C. We claim that if f = T; tr°then there is a unique bond of O'C leaving v.

Let the vertices of 2 2 in the face corresponding to v be a, b, c, d incyclic order, as in Figure 12 Note that f = ab, so a E C and b ELet I?, 22 - C - Ct. be the rest of 22 , i e, the set of vertices of 22 -Cin bounded components Note that, as C, is a component of 2 2 - C.there is no C,-R bond in 22.

Suppose first that d E C. Then e E C or c E since c is adjacentto b E C„, and so cannot lie in R. In the first case the bond ct is the

unique bond of 0"0 leaving v; in the second, de

I. Basic concepts and results

ti c

a E C b E Coc

Figure l2 The oriented dual bond f corresponding to au oriented bond ffrom C to Cr,:

Suppose next that c E C, Then d 0 R. so either dEemde C,

and again the claim holds.We may thus suppose that c C„, so c (which is adjacent to b E

lies in C, and that d C. If d E I? the claim again holds, leaving onlythe case a, c E C, b, d E C0 But C and C„ are disjoint connectedsubgraphs of Z 2 , so there are disjoint paths in Z-, one joining a to c, andone joining b to d As abed is a cycle in Z2 with no edges in its interior,both paths must lie in the exterior of this cycle, which is impossible; seeFigure 8.

As every edge in D= C' has a unique successor, the underlying lino/ i-ented graph D'C' contains a cycle S. This cycle separates the plane,and crosses only C,-Cm bonds; thus C., lies outside 5, and C is insideS. If f* E 0-"C with f = ob then, since one of a, b is in C and the otheriu C„ the cycle S must cut the bond f Hence f* E S In other words,ever y bond of DE C is in S. so ac consists of a single cycle q

In the rest of this chapter we present some basic results concerningcritical probabilities First, following Broadbent and Hammersley [1957]and Hanunersley [1957a; 1959], we show that the phenomenon of bondpercolation in V is non-trivial: the critical probabilities are neitheror 1. To do so, we consider the number p„ (A; r) of self-avoiding walksilk A starting at :r, where in graph terminology, a self-avoiding walk issimply a path All vertices of Z 2 are equivalent, and, as V is =!-iegulai,

pm = p.,(2?), pm(Z2;0) <4 x 3"-1

Lemma 2. Pot bond percolation in Z 2 we have

1/3 < PT < pu < 2/3

Piing As we noted ( "alien pr < in t in ' un: context:, so we roust showonly the outer inequalities

Suppose first that p < 1/3, and let Co be the open cluster of the or iginin the independent bond percolation' on 2: 2 where each bond is open withprobability p every site x E Cy., there is at least one open path from

to c Hence IC0 1 is at most: the number A' of open paths star ting at0 As a path in 52 of length 11 is open with probability p", we have

l im p" < 1 + 1 'OPYI <‘())= Ep(eoi) Ep(x)

Hence. pi > p.. Since p < 1/3 cv a s arbittat:\ it Follows that Pr > 1/3.

For the tipper hound we consider A = E2 together with its dual A' =± (1/2,1/2) defined ear hem taking a dual bond c- to be open if e is

closed, and vice versa An open dual cycle is a cycle in the dual graphconsisting of dual bonds that are open

Suppose now- dun p > 2/3. Let Lk be the line segment joining themight to the point (k,0), and let S be a dual cycle surrounding Lk of

length 21 Then .9 must contain' a dual bond c" crossing the positive

x-axis a some coordinate lat: Veen k 1/2 and (2( — 3)/2; thus we havefewer than t choices fin e' As the rest of S is a path of length 2( — 1in the dual lattice, win kIn is isomorphic to 2 2 , there are at most: f'p9m Ipossibilities fon S. Let Yk be the number of open dual cycles surroundingLk Since dual bonds are open independently with probability 1 — p.

c, -It( Pam — Pr _5_ -

9(3(1 — p))2(Eli( Yin <

As 3(1 — p) < 1, the final sum is convergent, so Ev (1"",.) —" 0 as k DO,

and there is some k with Ei,(4) < 1 Let il k be the event that Yk = 0;

since lE p Ork ) < I. we have 111 1,(A k ) > 0. Let Bc be the event that the kbonds in Lk are open Note that AR- and Bp a ye independent Also, ifboth hold, then there is no open dual cycle surrounding tire origin so,by Lemma 1, the open duster containing the or igin is infinite Hence,

0(p) = 00 (p) > P(Ak n By) = 1P,,(AMIP0 (80 = Pp (il k )ph; > 0

This shows that p H 2/3 wi5 arbitrary. ptr < 2/3 follows

The second put of t he argument: above, bounding the critical probabilityfrom above be estimating the number of separ at ing cycles, is sometimescalled a Peiells argument. after Pearls H9361. hi higher dimensions, we

>k+2 >k +2

I. Basil, concepts and results 17

estimate the number of separating surfaces: even the easiest bounds onthe number of these surfaces suffice to show that the critical probabilityis strictly less than 1.

Comparing Z d with Z2 and with the (2d)-regular tree, we see hat

1/(2d — 1) < p11' (DI ) < pllyZ(1 ) <2/3

As we shall see later, of these two trivial inequalities, the first is close toan equality

Tire bounds irr Lemma 2 can be improved by bounding pm horn aboveless crudely As noted by Hammer slev and Mm ton 119511, since p„, +„ <

/I n1 q„µ,,,, the sequence µ„ converges to a limit A = This li mi t isknown as the connective constant of V (see lirumnetsley 11957a1). Theproof of Lemma 2 shows that

I / A < ) ( 72 ) < 141(z2 ) < 1-1/A (1)

This was \\drat. Broadbent. and Hammersley [19571 and flanunersley119591 actually proved

In constructing a path step by step, at every step other than the firstthere are at most three possibilities, giving the bound A < 3 used aboveIn fact, looking three steps ahead, two of the 27 combinations conntedso far are always impossible, namely, those corresponding to turning leftthree times irr a row, or right three times It follows that A < 251/3Using a computer, it is easy to count paths of length 25, say, in a fewminutes. As 110 5 = 123,181,354,908,11/is gives A < 2 736

For p > 1/A, the expected number of open paths of length a in Z2starting at the or igin tends to infinity as n cx:. Thus, in analogy withnumerous phenomena in probabilistic combinatorics, one might: expectthat, with probability bounded away from 0, there are arbitrarily longopen paths starting at the origin This would suggest that = 1/A,as is indeed the case for the k-regular trees. The trouble is that, asthere are (A ± o(1)) 2 ” dual cycles of length 2n surrounding the origin(see Hantmersley (1961b1), the same intuition would indicate that forp < 1 — 1/A there are open dual cycles surrounding the origin, implyingthat p it = 1 —1/A. Thus, for both intuitions to be correct, A would haveto be 2 As we shall see now, this is riot the case

Any walk irr which every step goes up or to the right is self-avoiding.so p„ > 2" and A > 2. Let us say that a path P is a building block if Pstarts at 0, ends on the lbw .c +y = 2, and ever y inter mediate vertex ofP lies in the region 0 < .r g < 2: see Figure 13 Then any sequence ofbuilding blocks P I , Pr, , 'nay be concatenated to make a self-avoiding

18 Bask concepts and results

0

0 'a

Figure 13 A selection of building blocks that may be put together in anyorder to create a self-avoiding walk

walk IV star ting at 0. Star ting with If, each P i is the part of If lyingin the region 2k— 2 < 2+y < 2i (plus the vertex on 2:+y = 2i -- 2 whereII enters this region), translated to start at 0, so distinct, sequencesgive distinct walks.. Let ur„ be the number of walks If of length n thatmay be obtained in this way As there are four building blocks withtwo edges, and four with four edges, we have to. 1 > 4 2 +1 = 20, and

> // y .r„ > = 20 1', so A > 20 111 = 2 111 In fact., we have

ur n > 4w„- 2 -I- 4 Wu –.I

Solving the recurrence relation, it follows that A is at least the positiveroot of x1 — 4t2 — = 0, namely, 2.197....

Much research has been done on calculating y„ and bounding AFisher and Sykes [1959] showed that Ir i s = 17,215,332 They also ob-tained the hounds 2.5767 < A < 2.712, by considering respectively aspecial sub-class of paths, and walks with no short cycles More sophis-ticated algorithms and the use of computers have enabled these resultsto be greatly extended For example, / 5r was calculated by Conwayand Guttmann [1996] using a supercomputer (see also Guttmarm andConway [2001]), and Jensen [200‘la] has found fin The best publishedbounds on A are A < 2 6792, due to Penitz and Tittmann [2000], andA > 2.6256, obtained by Jensen [2001b] using the method of irreduciblebridges introduced by Mester, [1963]

Connective constants of other lattices have also been studied. Inparticular, writing Ad for the connective constant of Kesler/ [1961]

.1. Basic concepts and results 19

showed that Ad = 2d - 1 - 1/(2d) + 0(d- 2 ) More recently, HaraandSlade [1995] showed that

Ad = (2d) - 1 - (2d) -1 - 3(2d) -2 -16(2d) -3 - 102(2d) -I + 0(d-5),

and that a corresponding expansion exists to any order.

Returning to general graphs A, we next show that percolation is 'morelikely' in the bond model than the site model. In the arguments to conic.,we shall consider step by step explorations of the states of the sites, say.Suppose that the states X„ of the sites are independent, and each site nis open with probability p„. In each step, the next site v to he exploredwill depend only on the states of the previously explored sites Hence,given the history of the exploration so far, the conditional probabilitythat a is open is just N.

The reticle/ may well feel that the observation above needs no justi-fication. Note, however, that as a random variable, the histor y of theexploration up to step I depends on the states of all sites that might beexplored in the first I steps But the event that this history takes a par-ticular value (i e the event that, for 1 < s < 1, at step s we tested site asand found that X„ „ = i s ) is independent of X„ for any v { , to.}The next result is due to Hammersley [1961a]; certain special cases wereproved by Fisher [196

Theorem 3. Let A be a connected, infinite locally finite multi-graph.

Then

Pii(A) � Ail ( A ) and IP, (A) > (A)

Proof. Both inequalities follow from the assertion that, for ever y site xof A, ever y integer n > 1, and every probability 0 II)

(5)

Indeed, letting T1 0 it follows that

Or( A ) < P03(4) (6)

Thus, if 0,„(A lb,) = 0, then 0„,(Ap = 0, so Ai (A) > plA(A).

20

Basic concepts and sults

Also. if (5) holds, t

/IP\ 1,(1C,) = n

< EPLP!u=1

Hence, if tx„,(A ii) < ix, then x„,(N) < ,x, so p r (A) > (A)It remains to prove (5) As C .'„, is empty in At;, if x is closed, inequality

(5) is equivalent to

„(1C,4 > n 37 is open) (7)

In proving this, we may and shall replace A by the finite subgiaph A„of A induced by the vertices within distance 12 of X, since the event that

I C; I > n depends only on the states of sites or bonds in A„.Let us explore Ct.. the open cluster in the site percolation on A„ con-

taining conditioning throughout on being open We shall constructa random sequence 7 DI, U1 ) (.=1 of tripartitions of the vertex set1 7 (A„) of A„; this sequence will be such that the final set B1, obtainedafter a random number I of steps, will be the cluster The notationindicates that the sites in R t have been 'reached' by step t, those in DIare 'dead' (known to be closed).. and those in U, are 'untested'

To define T, set = Ur = 1" . (A„) \ {a'}. and D i = 0 Given(R t . D,,U1 ), if there is no [1,41, bond, set I = l and stop the sequenceOtherwise, pick a bond e t = yi z t \vith yt E E Utz and set (li n =

\ { Now test whetherr the site z t is open If so, set R1.4.1 =

Di= De. Otherw ise. set D14. 1 U }, 1? 1+1 = Bt . Note that. ateach step, the conditional probability that z t is open is p The processterminates as A„ is finite.

L3y construction, for ever is a connected set of open sites, andall sites in D, are closed Hence, as no site in Rc has a neighbour inUf = 1 7 (A„) \ (R, U the set Bt is precisely the open cluster Cs,.

To compare the distribution of rt,1 to that of let us explorein a similar manner, using a random sequence I' = 0 asabove This sequence 71 is constructed as T. except that, having pickede, = Th z t , we test whether the bond e t is open As this is the first (andonly) time we test e t , conditional on the sequence '7" , up to step theprobability that e t is open is just p. Consequently, the sequences '7- and

have the same distribution. In particular. 1C.7.1 = 1/41 and XI haveI he same distil ibution This implies (7) since Iry is contained in the open

Basic: concepts mid results 21

cluster C1.1! of ,r in A ll :: it. is the vertex set of the subgraph spanned by

the set of bonds we have tested in the process T' and found open. q

In tire argument above we could have worked directly in the infinite

graph, continuing the exploration indefinitely if (1;.. is infinite

One can also consider meted which sites are open with

probability p and bonds with probability p', independently of each other

Al r iting 0 2.(A; p, p') for the pi obabilth that there is an infinite path start-

ing at x all of whose sites and bonds are open, Haunnersley (1980) noted

that the more general inequality

pi ) < Os (A: p. rip')

for 0 < p, < 1 is air immediate consequence of its special case (6)

proved above

The inequalities in Theorem 3 need not be strict, as shown by the

k-regular tree. In most interesting examples, however, they are strict

Strict inequalities have been proved by Higuchi 119821, kesten [1984

Menshilwv [1987] and others For a general result that includes the

graphs r' as special cases, see Grimmett and Stacey [1998] Aizemnan

and Orlin/nett [1991] proved that a certain essential enhancement' of

a percolation model leads to strictly smaller critical values: under suit-

able conditions, adding edges to the graph strictly decreases the critical

probability This result has been extended la Beznidenhout, Griumrett

and Kesten [1993] and GI immett [199-1)

One Irriglit, expect that, if ) < 141 (\0), their ph(A ) < (An)

However, it is easy to construct examples to show that this is not tire

case, by modifying a graph in a way t hat decreases ph while leaving MIunchanged; see Wie g man 12003a1.

Our next result, due to Chinnuett and Stacey 119981, gives air inequal-

ity bounding A above by a function of p1 The proof will be a little

less pleasant than that of Theorem 3, clue to a minor complication intro-

duced to improve the exponent below from A to A — 1 Later, we shall

give a related result, for oriented percolation, where this complicationdoes not arise

Theorem 4. Let A he a connected, infinite graph with U1(13:i71111111 degreeA < x Then

141(A) C 1 — –Ph(A)Y1-'

(8)

The satire inequality holds I'm pi

9') Basic concepts and results

Proof It suffices to show that there is a constant It K(A) such that,for every site :r of A, every integer n > 1, and every 0 it is open) > ED/1i is(0 (1,1 > Ku), (9)

where r = 1 — (1 — We shall prove this with K = A + 1 Indoing so, we may. replace A by the finite submaph A,, induced by thesites within distance (A + On of x.

Let Cirb be the open cluster of :r in the bond percolation on A„ inwhich each edge is open independently with probability p, and let C.11:,be the open cluster in the site percolation in which x is open, and theother sites of A„ are open independently with probability ./ In order tocompare the distributions of 1(1,/11 and 1C1'„1, we first give an algorithmthat finds a significant fraction of C.1,14 and then show that a slight variantof tins algorithm finds a subset of C1:4,.

This time, we star t by exploring (4 1,4 using a random sequence T =Di ,(1i ) f of tripartitions of the set E(A„ ) of bonds. The notation

indicates that the bonds in L, are live' (known to be open), those in.D I are 'dead' (known to be closed), and those in Ur are 'untested' Ateach stage, the bonds of L, will for m the edge set of a tree containingr; we write Bi for the vertex set of this tree (Thus Id, is the set of sitesthat may be reached from x along paths consisting of bonds in L t .) Theset Ur will shrink as we proceed, while the sets L, and D, will grow.We shall also work with a set: Pt of si/cs growing with L namely theset Pt C V (A„ ) \ of 'peripheral' sites that will not contribute to thegrowth of li t The final sets Bur and P1, obtained after a random numbert of steps. will be such that R 1 C C R1 U P1

To define T. set tt, = E(A„), and D i = L 1 = 0 ThusGiven (L i , Di .dri ), let P, be the set of sites 'e 0 li t such that Ur containsa single bond flour Bi to v. If U, contains no bonds from Ri to 14 (A„ )\

U Pt ), set = t and stop tire sequence Otherwise, let e, = .//1 2: 1 E

be any such bond, with y, E ci Bi U P, Set U1 + 1 = Ur\ fedi,and test whether the bond e t is open. If so, set L,+ = U {e,} and

D, +1 = D,. so R i + = U {z,} Other wise, set Di + i = U {e,} andL 14. 1 = noting that lit+ , = A possible state of this explorationis shown in Figure 14 Since we never test the same bond twice, at eachstep tire conditional probability that e i is open is p. As A„ is finite, theprocess ternrinates

By construction, the bonds in L, are open, while those in D, areclosed. Thus Rt , the set of sites that may be reached from 4: alongpaths consisting of bonds in L 1 , is a subset of Ct..,1 Also, if c = yz is

1 Basic concepts and results 23

0 0

:10 4 •

0

ylt0 • O 0

6 O o 0

Figure 11 A possible value of D,, U,) when expiating the open cluster ofthe site re in bond percolation on 2S 2 Bonds in L,. DI and U1 are representedby solid, dashed and dotted lines, respectively. Tire set B1 is shown by filledcircles The site z lies in Pt , so the bond yz will never be tested; there is onlyone choice inzt for the next bond to lest

an open bond with y E Ri and z Ff .R1 . then c L i and e 0Di , soe E Since we stopped at step I. we must have E Pi , so the site z

is incident: with no other bonds of As z 0 Rt . the site z is incidentwith no bonds of Li . so all other bonds incident with z are in De, andhence closed

We have shown that all open bonds leaving 14 terminate in sitesthat me incident with no oldier open bonds Thus, every site z E C1,1;\Rtis adjacent to a site in and, crudely.

161;l1 5, (A ± 1)IRd (10)

We now turn to the site percolation on A„ in which 1 is open, and theother sites are open independently with probability r = (1 —To realize this probability measure, let XII y E V(A„ ) {t}, 1 < ( <

A — 1, be a set of i i d random variables. with P(X,, ,i = I) p andP(Xyi = 0) = 1 — p We decline y to be open if and only if at least oneof the X,, is equal to 1

Let its explote . a subset of as follows, using a tandom sequenceV = (LC, DC,Un ii _f of tripartitions of the set of bonds.. This sequenceV is constructed as T. except that, having picked e l = yt zi , we testone of the variables Mole precisely if this is the ith time that wehave chosen the satire site z, i.e if there are i — 1 values 1 < t withz s = z, then we test whether = 1 Note that i < A — 1: initially, z

Bas i c cancrids and results

IS incident v1/4-ith at most A untested bonds. each time we choose .7.„ zwe test au untested bond hidden!' with z, and we cannot choose zs

if there is only one such bond remaining Since each variable X,, istested at most once, each test succeeds with conditional probability p,

and the sequence has exactly the distribution of

By the construction of I', if e iyi z i was the bowl chosen at stept and e, E L b then the test at step t succeeded, so one of the X, isequal to 1, and z i is open. As m E it follows by induction onthat. Rr c Using (10) and the fact that R4 and 14, have the samedistribution, (9) follows, completing the proof q

For graphs A which are not regular, the proof above gives a slightlystronger result than Theorem 4, showing that if 0, > 0 Mr the bond per-colation on A in which each bond is open independently with probabilityp, then 0, > 0 for the site percolation in which the states of the sitesare independent, and the probability p„ that a site v is open satisfies1 - p„ = (1 - p)`r( " )-1 , where d(v) is the degree of r

The bound in (8) can be improved when pl.' i (A) is close to 1; in thiscase, the exponent A - I may be replaced by an exponent a little largerthan A/2 This is shown by the much mote difficult result of Chaves andSchomnann [2000] that. for any infinite connected graph A of maximumdegree A > 3,

1 AO) 5 1 92[4/21+1 (1 (A))LV2J

while there are such graphs with

/4i (A) > t - - pil'i(A))t4/2j

In prov ing (11). Chaves and Schonniann made use of the concept ofmixed percolation mentioned above. where states 11(:, assigned to thesites and bonds of A.

So far, we have considered percolation on ordinary graphs, in whichbonds are two-way. The basic concepts of percolation make just as goodsense for oriented graphs. where each edge is or iented horn one endpointto the other 'We could also consider threefed graphs, where there may bean edge in each direction between the same pair of vertices, or indeed,undirected or chiected In ulti- g raph s where there may be several edges(in one or both directions) between the same pair of vertices

In a I gated percolating the underl y ing (multi-)graph A is orientedWe assign states, open or closed, to the sites (vertices) or bonds (oriented

1)

tiN 7I \ >No

VV4 6VJ VV9

I Bosh: concepts and results 25

edges) as usual, and define the open subgraph, 01 –ATI„ as beforeNow, however, the open subgraph is of course oriented. An oriented

path is a (finite or infinite) path P =,rox i x, in A with edges zuhti+1oriented born x i to x i±i An open path in the oriented percolation isan oriented path in the open submaph, i e an oriented path all ofwhose sites (for oriented site percolation) or bonds (for oriented bondpercolation) are open.

Given a site', we write T..7 for the open out-cluster of x namely theset: of sites y reachable from by au open path:

C ty E A : there is an open path horn ,r to :111

C the open in–cluster of is the set of sites y nom ghichx can be reached, so y if and only if :u E CT,- Note that Cl"," is a

subglaph of the out-subtpstph A t of c!, i.e , the set of all sites y reachableborn x by oriented paths in A Since almost always we consider wily

CI, we write fig C' define 0, =01-, and x r ;\;:i as berme,using ea, = CC+.

As in the tutor iented case.. Or (p) and ;:v, (p) = are increas-ing functions of it, so we rues- define critical probabilities pn( ; ;g)andy-r( A ; JD as before Of course, all these quantities depend on whetherwe are considering site or bond percolation Sometimes we shall indi-cate this by writing tds (p), or plli ( A ;M t and so on In tlw oriented case,these critical probabilities may well depend on the site in consider, figexample, the tooted flee T shown in Figure 15, in which the root ...co has

Figwe IS An oriented rooted tree T in which = pc (1:y) pc(Mz).

two children, y and z.. all descendants of y have two children, and alldescendants of z have thiee children Orienting the edges away nom xyto obtain au oriented graph F. we have pc (T ; = 1/2 for y and all its

26 Basic concepts and iesults

descendants, and pc (i i ;:t) 1/3 for the remaining vet tices. where isany of li ' , p!I

For most graphs we shall consider, we do have pH (A ; a) = pr i (X; y)

for all sites x and y. and the same for pr; when this holds, we writepu(X) and p i (711- ) for the C0111/11011 values Indeed, if A is strongly

connected, i.e , for event panr x and y of sites there is an oriented pathhorn a to y, then, as in the 11110i iented case, it is immediate that 0„.(p)>0 if and only if 0 y (9) > 0, and v„(p) < cc if and only if x y (p) < (x. sopH and p t rue well defined

11 A is am, oriented graph obtained by or ienting the edges of a simplegraph A. then the paths in A are a subset of the paths in A Thuspercolation in A dominates that in A, so, for example,

j4( A;: r')�91(A)

for all sites X

The inequalities between site and ond critical probabilities presentedabove carry over to oriented percolation, with a slightly different formfor one of the bounds

Theorem 5. Let A be an infinite. connected locally finite oriented

multi-graph., and let a' be any site of A Then

( A 1./) /4 1 ( A : ,1') S 1 — ( t— A '''))s'n (12)

when ? ./..\ 1 „ is the 111(0')11111171 m. - degree of a cei te3 in A The same in-

equalities hold for

Proof Let pc denote eiabei pH or P with all occurrences of pc to beinterpreted in the saute way The first inequality. /OA; x) > p {I 1( A -,x),is proved in the same way as Them ear 3, mutat/5 mulandis: it sufficesto show that fia all sites .r, integers IL and 0 < p 1, we have

1Pfk. p(ICt n) 5_ p PI* p (ICI 1),

where here Cy.,, = Cit In showing this, we work in the finite subgraphA„ induced by the vertices reachable from by oriented paths in A oflength at most rn The construction of the sequences and I' proceedsexactly as in the proof of Theorem 3, except that we take the orientationinto auctorial: at each step t we select an edge ?TT = if-7-4( with ig e

a r E 1-1/.]'he proof of the second inequality, pg A; < — —pir!( A ; ./te'"

1 Basic concepts and ves tals 27

similar to, but simpler than. the proof of Theorem It; suffices to showthat lot evert' site x of A. every integer n > 1, and every 0 < p 1, we

have

, (iCrl x is open) Pi• p (tC;( 1 ?. a),

where r = 1 — (I — p) 411.. This time, we explore the open cluster CtIt) ofthe bond percolation containing in the natural way, using a tandemsequence 7- = (L t .M,Ul )i =1 of tripartitions of the set E( A „ ) of bonds,such that; the bonds of I4 form the edge set of a tree oriented awayfrom x. We write Br for the vet lex set of this tree; thus, Rt iti the setof sites that may he reached from .1: along ot hinted paths consisting ofbonds in Li

To define T. set U, = E( A „). and Dr = Lr = 0. Thus R, =fal Given (L t . D,. U,), if III contains no oriented bonds front It, to

V(X„) \ R t . set t I and stop the sequence. Otherwise, let 77 rrtbe any such bond, so Ut, E Br. and zi B., Set U1 4-1 = Ur\ 177 1,and test whether the bond e7 is open. If so, set Len = L, U {FI} andDi + 1 = pt. so ft = Rr U {z t } Other wise, set Db.. 1 D I URI and

= L i . noting that RNA = HI Since we never test the same hoodtwice, at each step the conditional probability that T1 is open is p AsA„ is finite, the process terminates By construction. the final set; R t isexact lv

Tfulning to the site percolation, we take i i d minion/ vatiables X„ t,

y E it ( A „) \ 1 < < A t „, with P(X„ = p and P(X„ = ()) =— p. and declare (r to he open if and only if at least one of the X„,i

is equal to 1 Thus the sites ir x a t e open independently, each withobribility r = I — (1 — We take a! to be always open As in tire

moot of Theorem we can use the K 11. 1 to construct a process Tv withthe same distribution as T Indeed, the definiticm is the same as that.of 7' except that, at step 1, having chosen 77 = Fri , we test whether

= 1, whew = : 1 < s < z s = zell As each 1.0 = if.,Ts isoriented towards z,, we have i < /..\;„ Tire rest of the proof is as before:the final set 14, is a subset of Cis.

The bound (12) in Theorem 5 is tight Mr oriented multi-graphs In-,deed, let A be an infinite binary tree with each edge oriented away fromthe toot Then MI (A) = = 1/2 (In other words, 41 (A ; d) =

pit'r(A;x) = 1/2 lot every site ,r ) Replacing each oriented edge by A.

parallel edges with the same calculation, we obtain a graph A1' 1 with

Basic emicepts and insults

maximum in-degree k, with

—401d ) = — ( W ) — /)1(7kI))k,

and the same for

If we do not allow malt ple edges, then (12) is still fairly tight when

Ai(A) A ) is close to 1; in particular, the correct exponent is indeed A i „ For

Mt(A) vent' close to 1, this can be shown using the oriented analogue ofthe construction of Chaves and Schonmann [2000] To get good boundsin a wider range scents to requite a atom complicated construction, whichwe now describe

By a (d, k)-mitt. we mean the oriented graph shown in Figure 16; thishas an initial site and a final site r The site v is the root of a 4-art'

Figure 16 A (3, k) wilt 'The initial site u s joined to all :3 k leaves of /I

oriented tree with toot II

tree of height k with ever y edge or ionted towards o (In the figure, thesolid lines show such a tree with k = 2 and d = 3 ) The initial site a

sends an edge to all (.11' leaves of this treeSuppose that the bonds of a (4, k)-unit me open independently with

probabilit p = \\'e shall take C > 2 to be a large constant, and let4

!pc Ignoring a for the moment, the numbers X; of sites at distance[tom n that are joined to v by an open path for in a certain branching

process: for I < i < A', is the sum of X i _ t independent copies of abinomial BUJ, p) Omaha) As d — x with C fixed, this binomial

dish Omaha/ converges to the Poisson distribution with mean C

.1 Basic concepts and results

continued to infinity, the till/ vivid probability of this branching processis a certain function p(C) of C.

We have p(C) = -c-c' -0 (CC- 2c ) as C — cc: the process is highlysupercritical, so given that it sin VIVOS at the first step, it is very likelyto survive forever. In fact, as k oc, with probability p(C) - 0(1) wehave XI; > Ckl" , say Let Rd, k, p) be the probability that there is anopen path horn v to v It follows that

Inn lim 1(1, k,C I d) p(C),d--cc k—x

so choosing d and then k large enough, we have f (d. k , C d) 1- (1 -C ±0 (ce-2c)

A similar but much simpler argument ha site percolation shows thatif a is open, and the other sites are open independently with probabilityp. then the probability g(d, p) that there is an open path hour v to v isbetween p and p - 0 ((1 - p) 2 ) for p close to 1, d > 3, and any k Let 0 <

< 1, and let T be a tree with critical probability pp(T) p 1 (T) = 7r.Choose an arbitrary site 3: 0 as the root, and or lent each edge of T awayhorn to to obtain an oriented tree T with maximum in-degree 1 whichalso has critical probability 1) 11 ( ; x0 ) = PT (7' ; ra) = (Depending

on the choice of T the critical probabilities MT; t) may or may notdepend on the site )

Let us replace each bond 5 of T by a (d. k)-unit, by identifying

it with 7 and v with y and then deleting Fa. The resulting graph Ahas bond and site critical probabilities p <1. 14( A ;10 ) = ( A ; to),

pc= /41 a0) = Tai

0) that satisfy

.1 01, 1,7, 10 = 7 and g(d, k, pr.) =

Rom the formulae above it follows that if ps, — 1, then 1 - "-1 –

Also, if d and k ;x are chosen suitably, then

G -1 ( 1 "1: + 0 (dpcb e -2dP:;) =

If plZ 0 and oc, this implies that

1 -11,! =(1_ pi? ) ( t--0(

It follows that, given any sequence p„ 1. we can construct_ graphs, -A„ with maximum in-degree Ai„(n) !N.; such that 74, =

/.4.1 ( 71 „ ; to) = 14 (71. „ ; to) and p,1; = pi.)[ (A „; :r0) = p4(7\ „; x„) satisfy

= psn = _(1_ /1, )( -I-0( ))Ai„

This shows that the constant Do, in the exponent in (12) is essentiallybest possible, at least tvhen 1

If A is an oriented graph and ir a site of A, then we have defined eightassociated critical probabilities: p H and pr for site and bond percolationon A and on the underlying simple graph A in any combination. Wehave eight trivial inequalities relating these probabilities; four of theform p r < pu, and four of the form pc (A) < pc (A;:r) Also, we havethat each critical probability for site percolation is at least that forthe corresponding bond percolation It follows_ that all eight criticalprobabilities are at least 'A (A) and at most p41 ( A ;3)

For an y d, let id denote the natural orientation of Z d , where eachedge is et iented in the positive direction Let p„ be one of A, ipl4/

Since all sites of Z d me equivalent, the critical probability pe ( aid ; ir) is

independent of a:, so we shall write p c,(2 .1) for the common value. Out

next: result shows that all eight critical probabilities associated to 12are non-trivial

Lemma 6,

1/3 < A (7,2 ) < /4 (E 2 ) < 80/81.

Proof We have shown that pli(272) > 1/3 in Lemma 2. It remains toshow that M 1 (2±, 2 ) < 80/81 The proof will be similar to the proof ofthe upper bound on ph(22 ) in Lemma 2 This time we shall use theconcept of a blocking cycle A cycle 5' in the lattice A' which is the dualof the unoriented lattice A = 2 2 is blocking if, for every oriented bond7.7 = oh of V with a inside S and h outside, the site h is closed; seeFigure 17

Fix p > S0/81, and let Co be the set of sites that may be reached fromthe origin by open paths in A where A = Suppose that Co isfinite. Since Co is a connected subglaph of 2,7 2 , by Lemma 1 the externalboundary O'Co of Co is a cycle in A'. If ab is a bond of 2 2 with ainside irc eo and b outside, then, by the definition of acc Co, we havea E Co and h r Co Hence, by the definition of Co, the site b is closed..Therefore, whenever Co is finite, there is a blocking cycle supoundingCo

Let .5' be any cycle of length 2t. As we walk mound 5' anticlockwise,each time we take a step upwards or to the left.. in order for S to heblocking. the site ar on (1 / 1 / right must be closed As 5' ends where

I Basic concepts and results 31

5 •

•

o 0 a o ••

Figure 17 A dual cycle S (dashed lines), and the oriented bonds from sitesinside 5' to sites outside S (arrows) S is blocking if every site pointed to isclosed

it star ts, we tabe as many steps upwards as downwards, mid as manysteps to the left as to the right. Thus we take a total of steps that goup or to the left. Any site x is on the right of at most one upwards stepand at most one leftwards step, so there is a set X of at least t/2 sitesthat roust be closed for 5' to be blocking Consequently

P(5' is blocking) G (1 — p)I/2

The rest of the argument is as in the unoriented case Let 1 k be theline segment joining the origin to the point (k, 0), and let I rk be thenumber of blocking cycles surrounding Lk We have already shown thatthere are at most tpoi_ i cycles of length 2t surrounding L k , and each isblocking with probability at roost (1 — p) (/2 so

EM)5 E (1121:-I(1— p)(/2 L, hlt-9 (3(1

C>k+2

As > 80/81, the final sum is finite. so 1EQ') —r 0 as k pc and thereis a k with E(1 20 < 1

Continuing as in the proof of Lemma 2, let A k be the event; thatYk. = 0, so P(A k ) > 0, and let Bk be the event that the k ± 1 sites inLk are open. Note that A i,. and Bk are independent Also, if both hold,then there is rro blocking cycle surrounding the origin, so c 0 is infiniteHence,

P(A)P(Bk) = P(ilk)pk+I > 0.> P(A k n 13k)

32 BOSic conceply and msalts

This shows that 4(172 ) /10/81 was at bit/ at ‘, p Ii r (2 2 ) <

80/81 follows. as desired q

Fin oriented bond peniolation, a cycle S of length 2( sin rounding the(nigh/ is blocking if even , bond (tossing S horn the inside to the outsideis closed As there ate exactly 1 such bonds, this event has pi obability(1 — p) t , and the atgamed- above shows that ph (2 2 ) < 8/9 Just asin the unor ionic(' case, one CM/ obtain better bounds by mote caudalcounting. Not every dual cycle surrounding the origin can arise as theextemal bou v d' Co of the open clustet Cy: ha example, the cycleS shown in Figure 17 cannot. as Hume is no WaV to reach the two sitesatt he bottom right along talented bonds inside S Balister, Bollobtisand Stacd, 110991 counted cycles that can arise as id 'Co mote carefully,showing that there ale (K o(1)) ( such cycles of length 2t, where K isa constant satisfying 5 1269 < h < 5 2623 The constant I = K(22)

plays one of the «Acts of the connectiveconstant A = A(2 2 ): the argumentabove shows that Ai' ( 2 2 ) < 1 — 1/K < 0 81. which is analogous to theImpel bound on di 'l (252 ) given in (-I) As we shall see in later chapters,much better bottncls on this critical pi @liability may be obtained in otherMINS

\VP IlOW bn it to the asymptotic behaviour of the vat hats et itical prob-abilities associated to percolation on E d and Z il As noted canter. foreach fixed (I, we have di nned eight et itical probabilities, of which pi; (Zit)

is the smallest and pC1 ( d ) the hugest. Since 2'1+i includes 2 4 as asubgt mill, each of these eight functions decreases with d.

Theorem 7. For any d > 2 toe have

Ph ( cid ) 5 14 (2:d)0(1/(1)2d — 1 —

Proof The fi t st inequality is just as easy as the special case d = 2 movedin Lemma. 2, Indeed, as 2( / is 2d-regular, the number p„ p„(4) ofpaths in Ed starting at 0 and having length n is at most 2d(24 —Fm bond percolation with p < 1/(2d — 1), the expected number of openpaths slatting at 0 is ) p„p" < so ;A(p) is finite and pp > p Asp < 1/(2d — 1) was arbitrary, pr > 1/(24 — 1) follows

Fat the tippet bound on 14 1 (2 a ) we shall in fact prove that

/4/(772))1/1(1/2'

l Basic concepts arid results 33

Note that the right-hand side is indeed 0(11d). as (Z 2 ) < 1 Since

M i (E d ) is decreasing. we may assume that d is exert Fix any po with

PH_1)2)2/4 so (1 _ pd)42 1),)Z < p2 1 and set pa = 1 —

We shall compare oriented site percolation on —E d with p = pd to thaton Z- with p = po It will be convenient to take the origin to be alwaysopen in both cases; this multiplies 00 (p) and \o(p) by a constant factor

1/1)By loget t of Ed , we shall mean the set of points whose coordinates

suet to 1, with each coordinate non-negative (In considering or ientedsite percolation starting at ft we may of course ignore sites with negativecoordinates ) Taking the origin open. let I? = Co fl y = t be theset of points of 72 in laver 1 that are reachable from 0 in the oriented sitepercolation on 2 2 Clearly . R i _ i , ever y site (r, y) itt layer 1 withat least one neighbour in is in RI with probability p, independentlyof the other sites in laver t. A site in layer t with no neighbours incannot be in RI By choice of ps, we have 00 (p9) > 0, so with positiveprobability every Ra is non-empty

Let y : — Z2 be the projection

//2

;ti,

1. I i=1/24-I

Thus is the linear map sending the first d/2 coordinate vectors toCI, 01 and t he last dI2 to (ft 1) Let the or igin of Ed be open., and eachother site be open independently with probability pd Let G be theopen cluster of the or iented site percolation on E d We will constructsequences (iro , WI , ) and (84 S i , SO, ) with the following prop-erties: for each point (a. y) E lit there is a unique V E ,91 such that

= 4, each 8, is a subset of Co, and (iro , .) has exactlythe distribution of (Ro, RA , )

We start the construction by taking So = R(') = {0} At each step inthe construction. R.', and S: will depend only on the states of sites inhirers up to l of Ed Given R1_, and Sr_ let us say that a point (44 iseligible if it is in layer t of Z2 and has at least one neighbour in R.;_ t Let(e, y) be art eligible point, and suppose without loss of generality that

— 1, y) E R.;_ I . Then there is a v E _ 1 with y(v) = (:r — Thepoint v has exactly d/2 neighbours win Zd with y(w) = y) Sinceeach is open independently with probability pd , the probability that atleast one is open is exactly pr If at least one w is open. include y)

Basic concepts mid results

in RC, and (one of the possible) w in SI Note that; we decide whetherto include (:c y) by looking at the states of its preimagcs under 5.-). Asthe sets of meirnages of distinct points are disjoint, each eligible (x, y) isincluded in .R; independently Thus, the distribution of Rrr conditionalon RC_ I is exactly that of R., conditional on R t_ i , and our constructionhas the required properties

As the distribution of (R,1„ ) is the same as that of (Ro, R h . .),there is a positive probability that every R; is non-empty. 13ut thenevery Si is non-empty, and Co is infinite, so 74 1 (2i < Ai , completingthe proof q

Numerous considerably stronger bounds have been proved about crit-ical probabilities in high dimensions. In particular, Cox and Durrett[1983] showed t hat

+ -1

+ o(4-") < pl]] (Y «/-1 +d-" + 0(4-1)

As we shall see in Chapter ‘1, .klensItikov [1986] proved that m i and firme equal in a very general context, including all the eases consideredhere, so we may write pr lot their common value Kesten [1990; 1991]showed that p]](I d ), (24)-1, and. independently, Hata andSlade [1990] and Cordon (1991] gave stronger results for p c/ ](Zd ). Con-cerning site per colation, 13ollobas and Kohayakawa [199-1] gave a simplecombinator ial argument showing that pr.(23d ) (1 + d"")-0)/(24)fact there are very precise formulae for both bond and site percolation:

1 7

p!!(Zil ) = — i 0(d- I)

24 4d-]1643was given by Nara and Slade [1995], confirming the first few terms in auexpansion repot ted in Gaunt and Buskin (19781 without rigorous errorbounds A similar expansion for site percolation,

31 75 +

5 +

9d SUP 324.]] 3914was given by Gaunt, Sykes and Buskin [f976] again without rigorouserror bounds

'Throughout this chapter, we have concentrated on the critical prob-abilities of independent percolation models As we remarked earlier,although these were the first graph invariants associated with percola-tion, there is likely to be only so much that can be said about themExcept in certain special cases, it seems that the exact critical proba-bilities arc in some sense rather 8/ hittair, Mr example, there may well

.1 Mt 51(1 concepts unit results 35

be no formula forpll(272), say, in which case tlfis quantity will never: be

kr /owl exact ly.The quintessential example of a known critical probability is the crit-

ical probability for bond percolation on Z2 ; we shall present this oele-

bt result of Han is and Kesten in Chapter 3. Using the same method,Kesten determined the critical probability for site percolation on the hi-

angular lattice Applying tricks of changing one graph into another, asin Figure 6, and the star-delta transformation to be discussed in Chap-ter 5, one way obtain certain othet critical probabilities in a similarway, including those for bond percolation on the triangular and hexag-onal lattices. Newt theless, these are only sporadic examples.

There is anodic/ , very different, percolation model whose critical prob-ability is known: face percolation on a random Voronoi tessellation inthe plane This result, is considerably Model than the results concern-ing exact critical probabilities on lattices: we shall sketch a proof inChapter 8

There are several other invar iants defined From the component slime-ture that me likely to be considerably mole significant, than the criticalprobability in the long term Rather little is known about these; weshall say a few words about them in Chapter 7

2

Probabilistic tools

In this chapter we present some fundamental tools horn combinatorialprobability that we shall use when we come to study percolation Theseresults have many applications in other alerts, for example, the stud y of

awful,'graphs.

One of the most basic results in paobabilitc theory i s oLi lam

Theorem 1. Let X = (X t , Kg. ) be a sequenm of independent ran-

dom variables and lel A be an event in the 47-field generated by XSuppose t, fin: every n the event A is independent of . X„Then P(A) is 0 a l 1 q

Note that different X; are not assumed to lace the same distributionAn event A with the proper ty described above is known as a tail event:

Theorem I states that fun tail ment in a product probability space hasprobability 0 or I

In fact. Theorem t was not kohnogomuLs original fthmulation of thisresult What he showed was that. if (Ai.-X2, ) is a sequence ofreal-valued uandontvariables. f : — P is Barre function and

IP( f (X) = 0, x,, ,x„) IP( .f(X)= 0),

then F(AX) = 0) is 0 or 1: see Kohnogorov B950, pp 09-70f AsKolmogorov noted, these assumptions ate satisfied if the Xr are inde-pendent, and the value of the [Unction f (X) remains unchanged whenonly a finite nurnbet of variables are changed

The following observation. known as Fekete's Lemma [19231 is fre-quently used to prone the convergence of various sequences

Probabilistic tools 37

Lemma 2, Let (a„) be a sequence of non-negative awls such that(I,,; < + aa, lar o. err > 1 Then lim„_, a„lir crisis

Proof Let e. li tn int „_, a„/(1. so 0 < e< or Given > 0 there is a

k such that a k < e + 11 -= kg + 0 < < then

a„ < ya k + < ga k +bk.

where bk = inax i < k a k aci Hence

a„ < ak / + bk la < c+ 2E,

if a is large enough q

If we do not assume that a„ > 0, then the conclusion is again that

a„la exists, but we must allow the limit to take the value

— 0c Taking login Ulm's, it follows that if (a„) is a snlmuthiplicative

sequence of positive /cal numbers, i e if a„+„, < a„a „, for I I, m >then ihri„_, a), /n exists

Many sequences occulting in percolation theory do not satisE N the

conditions of subadditivity or submultiplicativity dearth, and one needs

the following extension of Fekete's Lemma proved by de Bruiju and

Er dos [19521

Lemma 2'. Let Lt.) + Taff+ be an increasing Lunen° with

Ii :r(t)t -2 dt <rx.

and let (a„) be a segnenee of rents such that

a„ + „, < a„ + (1„, + veil + in) (1)

WhelleVCI u/2 < at < a. Then a„In --) L far some .L with x < L <

cc q

This result is essentially best possible For example, if ..r. > 0, then it

does not suffice to impose condition (1) for (1 + _)1112 < in <n

The framework fin much of combinatorial probability (including per-

colation theory). is the 'weighted hypereube (2 1”, We shall take our: time

over introducing this important concept

For any set 5. we may identify the power set -P(5'). i e , the set Of

all subsets of S. with the set {0,1} H of all functions f : S + 10.11 In

38 Probabilistic 10019

this identification, a set A C S corresponds to its characteristic function5' {ft 1}, defined by

{1 if y E A,14(r') =

0 if a: E S \ A

(We follow the standard convention in combinatmics that S C S )If S is finite (and this is the case we are mostly interested in) then S is

usually taken to be [id= {1,2, a}, and then 7(n) = P({/1]). 7(5)is identified with the hypercube simply, the cube Q" = 2" = {0, 1}",i e the set of all 0-1 sequences of length n Under this identification, aset .4 C [i] corresponds to the 0-1 sequence (a i );_ i in which 0 = 1 if

E A and a i = 0 if i [I]\ AIt is natural to consider Q" as a graph whose vertex set is 'P(a), in

which two sets A and B are joined if their symmetric difference AABconsists of one element Equivalently, Q" has vertex set {0,1}" Cand two eertices a and b are joined if a – b = :he; for some where(er,en, ,e„) is the canonical basis of T.' Thus the graph Q" has 2"vertices awl n2" –I edges

For A, BEP(n), put A < B if A C B. This turns the power set 7(n)into a partially ordered set, a poser. Equivalently, the graph Q" has anatural orientation: an edge ab is or iented from a to b if b –a = forsome i t 1 < r < a. Then a < b if there is an oriented path from a to b,i e., < for every i

A point (vertex) of Q" is naturally identified with the outcome of asequence of a coin tosses: for a = (a;),_r G (1 " we have a; = 1 if theith toss lands as heads This puts a probability measure on (tire vertexset of) Q". If we use fair coins then we get the uniform measure on Q",i e., the normalized counting measure: for A C Q" the probability of A

is 1P(A) = /2" If we use biased coins then we get a ' 11101C interesting'measure: if the probability that the ith coin lands as heads is p i , then

P(A) = H (i -pi) – (P7(1 10) Ho -m) (2)ft E = I air--(1

The cube Q" endowed with this probability measure is denoted by (>p,and called the 'weighted cube with probability p = (m)11_,

Putting it slightly more formally, let x i ,x„ be independent Bern-oulli random variables with Peri = 1) = p i and P(ari = 0) = 1 –Pi , andlet x = (:r i ) be the random sequence obtained in this way. Then Qp"is the space of these random sequences. Also, X = : = is a

2. Probabilistic tools 39

random subset of P(0), so Q v" is the space of 10,7140111 subsets of [nl, inwhich the elements 1,2, , 0 are chosen independently of each other,and i is chosen with probability PaIf = 1/2 for every then the weighted cube is just the unweighted

cube mentioned above; in this case the probability measure is the nor-malized counting measure In the next simplest case we have pi p forevery i, so that P(X = A) = 9 1 A1 (1 – p)" – F AI ; in this case we write (21';instead of Q p" The standard percolation models correspond to (2;,'

The events in our probability space (2` me the subsets of Qpn , i e ,the subsets of Q". A subset U C Q" is a (monotone) Increasing event,01 simply an up-set., if a, b E Q", a E U and a < b imply that b E UReplacing Q" by P(S), a set system U C P(S) is increasing, or an up-set, if A C B C X and A E U imply that B EU D C (2" is a(monotone) decreasing event or a down-set if a,b E (2", a E D and a> bimply that b E D Equivalently, D C P(X) is a decreasing set systemor a down-set if A C B E D implies that A E Clearly, U C (2"is monotone increasing if and only if its complement, .D = Q" \II, ismonotone decreasing

The fundamental correlation inequality in the cube, proved by Harris[1960) in the context of percolation and rediscovered by Meitman [1964states the intuitively obvious fact that increasing events are positivelycol related.

Given a set A C (2", for t = 0,1 define

= {(ai)72 1 1 : (al, ,a„)) 1 ,1)E C (2"

II A is monotone increasing then Ao C A 1 , and if A is monotone de-creasing then Ar C Ao Let Q p",–I be the weighted cube whose proba-bility measure is induced by the sequence = (m)72 / 1 . With a slight(and usual) abuse of notation, let us write P for two different measures,namely the probability measures in Q p" and (47 1 Note that

P( A ) = ( 1 – )P(-410) +Th iP ( A r) (3)

for every set; A C Q"We are ready to state and prove Harris's Lemma

Lemma 3. Let A and B be subsets of Q p" If both are up-sets or bothale down-sets then

P(A n B) > P(A)P(B). (4)

-10 Probabilistic tools

If A is 01) up-set and 13 is a down.-set thco

P(A n B)< INA)P(B)

(t)

Proof Let us prove (1) by induction on II Pot n = I ((a, indeed, a = 0)the inequality is ti kris'. so suppose that I] > 2 and that (l) holds for

— ISuppose that A and B ale both up-sets <A both down-sets Then

either Ao C At and Bt C or else A t C Ao and C Bo hipal t iculat

OP(Ao) — (P(Bn) — 12(13t))? 0(6)

Also. by (3), the induction hypothesis and (6) we have

P(A n B) =(1 –p,JP(A,nBn)-I- p„IP(A, n B,)

— PiP( A o) P(Bo) -f-PaINA1)111(B){ (1 — )P(lo) N IN A / — Ps)P(B0) p„P(13

= P(A)P(B).

with the second inequality following how (6), and the last equality aconsequence of (3)

It01 A On up-set and 13 a down-set. inequalit y (5) follows IA applying(1) to the up-sets A and .13` (2” \ 13:

P(A B) = P(t) — P(.4 n Be)

< P(A) — P(I)P(Be)

= P(.4) — P(A){1 — P(B)}

= P(A)P(B) q

The extension to Minim product spaces is iminediat we shall not needit lane

As the intersection of two up-sets is again an up-set Lemma :3 impliesthat

P(.41 n .42 n n A i ) > P(Ai)P(A2) P(A)) (7)

whenever A I . me all up-sets, °I all down-setsA simple consequence of Hailis's Lemma is that if At, rue

increasing events in (2 11,2 whose union A has very high probability. thenone of the A; must hate high probability Indeed, the complements Ay

are decreasing, or down-sets, so from (7) we have

H N I ") .5- NA')1=1

It follows that fot some i we have

P O) 5- (11)(Ac))11?

IP(*) I — — PO I U U AO)

(8)

In the case where each A; has the same probability.. inequality (8) holdsCot ever) i. For t = 2, this observation is sometimes known as the'square-root trick; for t = this is the ''nth-toot trick'

If p : Q" R is any function, then we may think of II as a signedmeasure on Q" Thus, For E C Qn we have

p(E).= Fr(x).

and the integral of a function h : Q" — R with respect to p is

h ip= h(3)/1(3) = (HO(Q")EQ.

In pat ticultu, wining l f for the diameter istic function of a set E Cwe have

It^ dt = p(E)

In this notation. Harris's Lemma states t hat for the probabilit.v measurePp on the cube (.2 p" we have

II1.A1.13 (Pp > A drinp Li dIPp(9)

whenever A. B C Q" me up-setsInequality (9) yields a more high-blow formulation of Harr is's Lenarta

Although the terminology is self-explanatory, we note that a functionIt : Q" R is (monotone) increasing if h(r) G h(g) whenever < y

Lemma 4. Let f and q be increasing functions on Q" Then

I f g dPp > ()Pp g drp(10)

42

Probalribstie tools

Pivot Adding a constant C to f increases both sides of (10) b y the

same a/1101111i, C' g As C f is positive for some C', we may

thus assume that f > Similarly, we may assume that q > 0 Then= rift and g = j= where the ef and di are positive

constants and the functions h, gi are characteristic functions of up-sets

Thus.

I Ill cid; Lg.) c i d; L gjI 11,

where the inequality is Lemma 3 q

The van den Berg-Kesten inequality I1985I fin monotone events ispartial converse of Harris's Lemma: it also states an intuitively obviousfact Let .4 C (2; be the ineteasing event of having at least one inn ofthree heads, and let B C Qp" be the increasing event that there ate at

least five heads Let AO /3 be the event that there is a tun of three heads

and there ate at least five Ow/ heads It is Mud not to he convinced

that IP(A 0 B) < INA)P(B)Let us give two locum definitions fin the 'square' or b operation

for set systems First, we define AO B fin B C P(.5)

A 0 C S thole ate disjoint sets S such that

,--cnv implies D E A. and

Dnz=cnz c B. fin any D C S}

Let us write / I x for the restriction of a function to a domain X If

= d = ((.1 [ )i 1 . and I C [al. then the condition c hi = means

exactly that = to t all i E I For A, B C Q" = {0,1}", we may

define A D B by

A q B e (2' thew are disjoint sets 1 ./ C [a] such that

d ii = e ll implies d E A, and

= / implies d E B. for any d E C"2"}

This definition is equivalent to that fin set-systems given aboveNote that A0 B is a subset of A ft /3 In fact, it may be a ratlar small

subset of A n 13: fin example, if neither A not B contains a subenbe of

dimension at least 42. then A q B 0di/ = q 1 implies that d E A, then we call c ti a witness on a cc/iificate

161 A with suppoll 1 Thus AUB is the set of points c fin which there aledisjoint sets I and .1 stab that c it is a witness lot A and e l [ is a witness


tht B Note that the square operation, which is obviously commutative,is also associative: both (A l 0 AO 0.4 3 and .4 1 0 (A9 0 A3 ) consist ofthe set of points c for which there ate disjoint I I , 1) and 13 such thatel]; is a witness for A. = 1,2,3

For increasing events (and for decreasing events), the definition of thesquare operation can be simplified considerably Indeed, if A, B C P(S)are hICI easing set systems, then

A D B = {21 UB:An13 (4, A E A, B E B}

Lt the context of inmeasing set-systems, we may take a witness for A tobe simply a set A E A. /abet than the function that is I on .4 ThenA 0 B is the set of sets C such that. C contains disjoint witnesses lot Aand Mt B

identifying (2" = {0,0" with the algebra 272{, so that for a =(a i )7=1 and b (1)) )7_, we have = (a + 14)7- 1 and oh = (oihi);1-1,with 1 1 defined to be (I, we have I he following simple description ofA 0 B for increasing subsets A, B of Q":

A 0 B = b: ob = 0, E b e 131

In other words, for increasing events A and B C (2", the event A0 Bhappens if sonic (minimal) elements of A and B occur . disjointly: A OBs the up-set generated by disjoint ly supported elements of A and B. as

in the example we star tied out with.Ebb then, is the van den Berg-I:08ton inequality The simple proof

below is due to Bollokis and Leader

Theorem 5. Let A, B C Vr; be increasing events Then

P(.4 0 B) C P(A)1P(B) (11)

Proof We shall prove (11) by induction on n; the case it 1 (or,indeed, the case n = (1 which makes perfect: sense) is trivial

As before, let Q p"7 I be the weighted (n - 1)-dimensional cube withprobability measure defined k = ,p„_,) Let C = A0 B Itis easily checked that, as A and B me up-sets,

Co = Ao Bo

and

C', = (.4 0 0 U (A I 0 DO (12)

41 Probabilistic tools

As A0 C A, and Bo C it follows t hat

Co C (.4 0 0 ) n (A 1 0 B0)

(13)

and that

C A Bi

Thus, by induction,

P ( C0) = P1A0 0130) P(Ar)IP(BoT

and

P(C, ) < F(.4, q B,) < P( )

Furtheinnne, front (12) and (13).

F(Co) INGO < LP ( (,4 0 0 B I ) n ( A 0 Bon

+P((.40 q /3 1 )1T (A, q 130))

P(A 0 0 B 1 ) + PHA 0 B0)

< IF ( :1 0) IPTB D F( A , )Fd131)

Multiplying the last three inequalities b y (1 — p„) 2 pY, and p„ (1 — p„)

tespectivel y and summing. we find that

(1 -1,,,)P(co) Puflet

{ — POP( Ain + Pit P ( T t )1 — fin ) 121 Bol +PaLNBI/}

Using (3) three times, this is just) < P(A)P(B), as required q

Van den Berg and kesten [1985] cot/lectured that their inequalityholds lot all events, not only monotone events Van den Berg and Fiebig[1987] proved that the inequality holds lot Coll Vel events, that is, inter-sections of increasing and decreasing events Tate lull conjecture resistedall attempts until Rchnet [2000] moved it

Theorem 6. /A A B c QV, Then P(±1 q B) < P(A)P(B) q

The proof is much hander than that or the van den Berg—Nester inequal-ity

Han is's Lemma has many extensions, including some cal 'elation in-equalities of thiffiths (1967a:1967bl concerning Ising lenomagners, andextensions by Kelly and Shetman [1968] The PEG inequality of Ra-tuin, kastelevn and Ginibre [1971] extends these to a cot telation inequal-ity On pat tinily °Rioted sets. with applications to statistical mechanics;

9. .Probabilistie tools 45

the EKG inequality is the roost often quoted correlation inequality inphysics: even Hair is's Lemma tends to be called the 'EKG inequality'

Ahlswecle and Davkin (19781 extended the EKG inequality to a verygeneral correlation inequality on lattices What is amazing is that suchan inequality could be true: its proof, although fa.r flour trivial, is not

ver y difficult.Given points a = (at)(_, and b (b 1 )7_, of the cub e Q" = {0,1}",

their jour is a V b = (ei ) iL l , where ei = a i V h; = maxfa i ,bi l, andtheir meet is a A b = (d1 )7_,, where d i = a; A b i = nrinfa i ,b 1 1 (Thus,identifying (2" with the power set 'POrd), so that a and b become subsetsof H. we have aVb=aUb and a Ab=anb) For A. B C Q". define

the join of A and B as

A V 13 = la V b: a E_-1. b E B}

and their nee/ as

A A B = (a A b:a EA. b E B}

As usual, we shall identify a function * Q" = (0, II" IR with thesigned measure it defines on Q". so that if E C Q" then

= =)((')cEE

Here, then, is the Foul Functions Diemen' of Ahlswede and Dar kin,stating that a certain trivial necessary condition for an inequality is alsosufficient

Theorem 7. Let o. Q" — Pr'` = (0. yr) be such that

n(0)/1(b) < 1(a V b)6(a A h)

for all a b E Q" Then

el(A)/3(B) G 2(A V .8)6(A A B)

fat all subsets A. B C Q"

Choosing appropriate functions o. 3, 2 and 6, Theorem 7 iruplieshost of inequalities. In roost of these results, a, (3, 2: and 6 ate chosento be the same function (measure) p: Q" IR T We may choose any

function p, provided it is log-supermodulm, i e satisfies

p(a)p(b) p(0 V b)p(a A b)

0

16 Probabilistic tools

for all a b e Q".. (Occasionally, such a function p is said to be log-monotone ) By the Ahlswede-Daykin Four Functions Theorem, if p is

log-super. modular , then

p(A)p(B) C p(A V B)p(A A B) (14)

for all A, B C Qn . The very special case where ft is the normalizedcounting measure implies Harris's Lemma, since, when A and B areincreasing, A V B is just A fl B The simple proof of Lemma 4 shows

that (14) implies the FIKG inequality, which may be stated as follows.

Theorem 8. Let p : Q" P. ± be a log-supermodular probability mea-sure, and let f. g: (2" --., F', + be increasing functions Then

fg dit � 1 1(11!!gdtr

Our final aim in this chapter is to present, some fundamental resultsconcerning thresholds for events, stating that under rather general con-ditions, the probability Pp (A) of an increasing event A C 1111C/C1 goesa sharp transition as p passes through a critical value These sharp-threshold results will be of fundamental importance in several of themajor results on percolation.

Let A be an event in the weighted cube Qp" Given a point w E

Q" = the it II variable w i is pivotal for A if precisely one ofw= (uo, . co„) and i(w) = . — wi+1 , , w„) is in ANote that: whether the dIr coordinate is pivotal depends on the point wanal on the event A The influence of the ith variable on A is

di (A) = iip (A) = Pp({w: a); is pivotal for AD,

or, in the usual notation of probability theory, simply

/3i (A) = Pp (w i is pivotal for A)

The fundamental lemma of Margolis [1974] connects the derivatives ofPp (A) with the influences of the variables; this lemma was rediscoveredsome years later by Russo (19811.

Lemma 9. Let A be an increasing event in, the weighted cube Q p", wherep = ( p i• , p„) Then

p; Pp ( A ) = /51(A)

and

and note that


In particular,

dlp r ( fl ) = /3r (A).

Proof It suffices to prove the first statement with i = a Given a pointx = ,:rk) E Q k , where k < n, write

= f Pi H - Pi)i: =0

so that if k = n then Pp ({x}) = px Also, for x E Q"- 1 , set x+ =(x ) , , 1) and x_ = (r h . -1,0), so that xi., E 2" Notethat px+ + px - =

For an up-set A C Q", let

A„ IxE(2" -1 x+ E A, x E AI

fi b z= E (2 "-I x+ E A, A},

Pp( A) = E (px Px-)

E px -}- Pm )11 Px

(15)xEslb

Hence

pl? (A)Op„

At the point x' = (x,x„), the nth coordinate is pivotal if and only ifx E A i„ so this last expression is exactly /3„(A) q

To prove sharp-threshold results, we wish to find lower bounds onO i (A) In the unweighted cube Q„, /31(A)2"-1 is precisely the

edge-boundary of A Thus the edge-isoperimetrie inequality in Q„ tellsus that if P 1t1A) f then

11

0 1 (A) > t2"(n logo (t2"))/2" -I = 2t;log2(1/t),

SO

Px

max/31(A) 2t log; (11t)In

S Probabilistic tools

Ben-Or and Linial {1985: 1990] conjectured that this last inequalitycan be improved substantially: up to a constant factor, logn(1/t) canbe replaced by logo 'this conjecture was proved by Kahn, Dalai andLinial 11988] with the aid of the Bonarni-Becknea inequality from har-monic analysis For a combinatorial proof, see Falik and Samorodnit-sky [2005]

Theorem 10. Let A be a subset of the n-dimensional discrete cube

(2„ = {0,1}" with probability t = 1A1/2". Then

fi i (A) 2 > et2 (1 _ ( log 02 (16)11

where > 0 hi 1171 absolute constant q

The bound above is often written with min{ t 2 , (1 — 0 2 } instead oft 2 (1— 02 ; apar t Rom a change in the constant, this makes no differenceA similar comment: applies to the bounds below

Relation (16) immediately implies that

max > t) (log n)In

Ben-Or and Linial gave an example showing that this is best possibleup to an absolute constant

In order to apply Lemma 9, we need bounds on influences in weightedcubes Such an extension of Theorem 10 was first proved by Dour gain.Kahn Kalai, Katznelson and Linial [1992]; a simpler proof has sincebeen given by Fr ieclgut [2004]

Theorem 11. Let A be a subset of the rueiglrted cube (2 11; with probability

Pp (A) = I Then

!off nmax/j; (A) > d(1. —

where c > 0 an absolute CO 11 stant q

Friedgut and Kaki [1996] noticed that a slight variant of the proofof Theorem 11 gives a stronger Jesuit (apart t from the constant): if themaximal influence is not much larger than the bound in Theorem 11then there are many variables of comparably large influence, so the stunof the influences is huge

Theorem 12. Let. .4 be a subset, al the weig hied cube Q" with Pp (A) =

If 11;(.4)< S fat every i. then

,di (A) > c — ) log(1/6),i.1

where c > 0 is an absolute constant q

There is a simple condition uncle/ which one large influence guar anteesthat the sum of the influences is large A set C Q'' is signmettic ift is • wariant under the action of sonic group acting transitively on[n] Thus, A C Q" is symmetric if, for all 1 < j < h < n, there is apet mutation rr of [n] such that 7(j) = h and if x = CIA is in A thenso is 7 (x) = (.Ho)i All we shall need about a symmetric event is thatever y variable has the same influence on it

Using this observation and Lemma 9. Ft iedgut and Kahl [1996] de-duced from Theorem 11 that ever y svn unetnc increasing property ifihas a sharp threshold: an 0(1/ logo) increase in the probability p suf-fices to increase the probability of the property from close to 0 to closeto 1

Theorem 13, There is an absolute constant c i such that if .1 C Q" issymmetric and increasing. 0 < z < 1/2, and IPp (.4) > then Pg (51) >1 — e Whell CM]

log(1/(25)) — p >

log o q

1'riedgut and Kalai [1996] showed that, for events whose thresholdoccurs at very small values of p. this result can be strengthened consid-erably

Theorem 14. There is an absolute wnstant c, such that if rl C Q" issymmetric and increasing. 0 < e < 1/2, and Pp (A) > then TVA) >

— e whenever

loo(1/(25)) — e2P (1//) log

It is not: hard to adapt the proof of this result to powers of probabilityspaces with 3, 4, 5 elements, where all elements but one have smallprobability

0

3

Bond percolation on Z2 - the Harris--KestenTheorem

From the publication ' of the first papers on percolation theory in thelate 1950s, for over two decades one of the main challenges of the them)was the rigorous determination of p it = the critical probability-for bond percolation on the square lattice. Hammersley's Monte Carloexperiments suggested that the value of p H might be 1/2; [rather ev-idence for this was Own by Bomb (1959], Elliott., Heap, Morgan andRushbrooke [19501, and Dumb and Sykes (1961]

The fist major result on this topic was due to Harris [1960], whit)moved that p H > 1/2 In the light of this result: and the Monte Carloevidence, Hammersley conjectured that the critical probability is in-deed 1/2. Sykes and Essam [NMI] gave a non-rigorous justification of

( 4.72 ) = 1/2, fur ther supporting this conjecture.the next important step towards proving the conjecture was taken

by Russo (INS] and by Seymour and \Wish [1978] who, independen(t)s

proved that A (Z,12 )+ gl ij (Z2 ) = Kesten [1980] finally settled theconjecture: building on the Russo-Seymour--Welsh result, he gave aningenious and intricate proof that. p H = 1/2.

By now, there are many proofs of this famous Harris-Kesten Theorem;in fact, there are a number of global strategies based on various differ-ent ingredients, and frequently even the same ingredients have sever aldifferent. proolS. :Here we shall give several variants of what is essen-tially one proof, using some of the basic probabilistic results presentedin Chapter 2 NA, T hat follows will be heavily based on the presentationin Bollobtis and Riordan [2006c] Later, in Chapter 5, we shall indicateanother proof, based on Menshikov's exponential decay theorem and theuniqueness theorem of Aizemmm, Kesten and Newman

Intuitively, the main 'reason why' p it 1/2 is the fact that, for p =1/2, the probability that there is an open path crossing an n by n —

Band percolation on Z2 - the Hann—Hester'. Theorem 51

rectangle the 'long way' is 1/2 As we shall now see, this is an immediateconsequence of self-duality However, haying proved this simple fact, weale still lather far from proving that pu = 1/2: in some sense, the realprobleru struts only then

Recall that the dual of A V is the lattice A* with vertex set {(a -F

1/2, b 1/2) : (a,b) E Z2 1, in which sites at distance 1 ate adjacentThus there is one dual bond e* for each bond e of V; this bond is thebond of A* that crosses e

A rectangle R in V is a subgraph induced by a set of sites of the form[0 ,14 x where a < b and e < d are integer's We shall use the samenotation for the vertex set [a, x dI and for the rectangle it induces.If b— a +1 and I'. = — 1 then we call R a k by C rectangle; notethat such a rectangle has 0 sites and 2k1 — k — C bonds. A rectanglein A* is defined as in A = V; equivalently, R.' is a 'octangle in A* ifP,.`+ (1/2, 1/2) is a 'octangle in A

Although we have defined rectangles as suligiaphs of V and its dual,it is natural to define an 'abstract' k by C rectangle as a 'grid graph' withhe vertices and 2/cf. — k — (edges. Cleat ly, a suligraph of 23 2 isomorphicto a k by C abstract rectangle with k, C > 2 is a k by C rectangle asdefined above

Turning to bond percolation on A = Z 2 , let us consider a configura-tion w E {0,1} E(A) on A. For the moment:, the measure on the space ofconfigurations will be inelevant, lmt, we shall still call a set of configu-rations an event As in Chapter 1, we define a bond e" inr A" to be openif e is closed and vice versa; thus the configuration w specifies the statesof the bonds in A and in A*

The hot izontal dual of a rectangle R = [a, x [c, di in A or A' isthe rectangle Rh = [a + 1/2, b — 1/2] x [c — 1/2, d + 1/2] in the duallattice Analogously, the tun beat dual of R is the rectangle R" = [a

x [e+ 1/2, d — 1/2]; see Figure I. Somewhat: artificially, thehorizontal dual of a 1 by l rectangle is the 'empty rectangle', as is thevertical dual of a k by 1 rectangle. For k, f > 2, the horizontal dual ofa k by C. rectangle is a — 1 by C + 1 rectangle, and the vertical dual isa k + 1 by C — 1 rectangle Also, (Rh )" (R'') h R.

Given a configuration ro, an open horizontal erossini«A a rectangle= [a, x [c, 4] in A or A' is an open path P C R joining a site (my)

to a site (I), 2), i e., a path in the appropriate lattice, all of whose edgesare open., joining the left-hand side of R to the right-hand side. We write11(R) for the event that R has such a crossing Similarly, we write 17(R)

for the event that I? has an open. VC/ heat crossing, defined analogously.

52 Bond ingeolaboa O n Z -- the flail Theorem

MIMISWIM111111111111111111111Figure 1 A rectangle 1? (solid lit and its horizontal dual Rh (dashed linRh is the Nettie& dual of B

As a minimal open houzontal crossing of a rectangle I? does not con-

tain anv bond joining two vet flees on the same vertical side (left or right)

of I?, the event fi(R) depends only on the states of those bonds in I?

whose duals appear in Rh Similarly V(R) depends only on the states

of bonds with duals in RE.

The next lemma is the pit ()wised 'reason why' pn (22 ) = 1/2 The re-sult is obvious, and it is tempting to state it without a proof However.

it is not entirely t t ivial to pr ove, and it usually takes quite a while to clot

the i's; for a poof of it closely /elated result see Kesten [1982. pp 386-

:3021 Here, we shall give a simple proof needing no topology Logically.the proof is equivalent to that given in Bollobas and Riordan [2006cdbut the presentation is mote like that of a related result in Bollobas and

Riordan [20061].

Lemma 1. Let I? be a rectangle in E li of its dual. Whatever the states

of the bonds in B. entelly one of the events 11(R) MI (1 V(R h ) holds

Proof .. Consider the partial tiling of the plane by octagons and squares

shown in Figure 2.. (This is, in fact, part of the Atchinuidean lattice

8-); see Figure 18 of Chapter 5 ) We take a black octagon for each

site of R. and a white one for each site of Rh The bonds of R and

of Rh ate represented ha squares. with the sante square representing a

bond e and its dual e* A square representing a bond e of R and its

dual C " in H" is coloured black if c is open. so g* is closed, and white

if c is closed and e* is open In the first case„ the black square joins

Bond petrolation on Z2Harris lteodem 53

Figure 2 The upper figure shows the open bonds of a rectangle B (solid) andits horizontal dual Rh (dashed) In the lower figure each site of I? is drawn asa black (shaded) octagon, and each site of R h as a white octagon The centralsquares correspond to bonds e E I? with duals e' C R fi ; such a square is blackif c is open and white if e* is open There are additional black/white squaresaround the edges to connect; the sides of the rectangles

the two black octagons corresponding to the sites e i0111,9 In the second,

this white square joins the two white octagons corresponding to sites of

54 Bond pocalgtion on 27,2 - the Hartis-Kesten Theorem

the dual lattice that e" joins The squares corresponding to the bondsin the vertical sides of I? me coloured black, to 'join up' each side ofR; the states of the corresponding bonds ate irrelevant to the event11(R) Similarly, the squares corresponding to the (dual) bonds in thehot izontal sides of Rh ale coloured white

Note that II(R) holds if and only if there is a black path of squaresand octagons from the left of the figure to the right, and 1 1 (1? ),) if andonly if there is a white path of squares and octagons front top to bottom.In particular, 11(1?) and V(Rh ) cannot both hold: otherwise, these blackand white paths contain disjoint (piecewise linear) curves in the planelying in the Ulterior of a cycle C (the boundary of the partial tiling),and joining two pairs of points at, and bd, with a. b. c and d appearingin this cyclic order around C As noted in Chaplet 1 (see Figure 8), Kr,could then be drawn in the plane

Let I be the interface graph. formed by taking those edges of oc-tagons/squares that separate a black region front a (bounded) whiteone, with the endpoints of these edges as the vertices Then every ver-tex of I has degree exactly 2 except for the four vertices x, y, z and at.which have degree 1. Thus the component of I containing a" is a path Ili,ending at another vertex of degree I. Walking along I1 7 front r, there isalways a black region on the right and a white one on tire left. Thus 147cannot end at: 4, so IV ends either at y or at

If II' ends at y, as in Figure 2, Own the black squares and octagonson the tight of IV give an open horizontal crossing of I?. More precisely,these squares and octagons correspond to a connected subgraph S of Rjoining the left. of .1? to the right, all of whose bonds ate open, exceptpossibly lot some vertical bonds in the sides of I? Let P be a minimalconnected subgtaph of 5' connecting the left of I? to the right Then Pis a path. and P uses no vertical bonds in the sides of B. so P is anopen horizontal crossing of R. and 11(R) holds Similarly, if IF ends atw, then the white squares and octagons on the left of II' give an openvertical crossing of R h in the dual lattice. Thus at least one of I1(R)and V(Ie) holds. q

The path II' in the interface graph I described above may be foundstep 1w step: we enter the tiling at r, and at each vertex we 'test' thetwo edges leaving tins vet tex and follow one of them. If II(R) holds, sot hat IV leaves the tiling at y, then the path P we find is the the top-mostopen hot )contal crossing of I? This has the very useful property that itcan be found without examining t he states of bonds below P

Bond percolation on Z2 - the Hands-Kesten Theorem 55

Algorithms such as this, that test for crossings by examining 'inter-faces', are sometimes of practical significance, as they can be much fasterthan exploring, say, the set of all vet tices of R reachable from the left,and testing whether this set contains a vet tex on the tight

Lemma 1 may be worded as a statement about a plane graph andits dual: contract each vertical side of B to a single vertex, and add asingle edge I joining these vertices to form a graph G. Also, contracteach horizontal side of Rh to a single vet tex, and add an edge f* joiningthese vet tices, obtaining a graph 6* Then (.7 and G* are planar duals;see Figure 3

Figure 3 A rectangle /? with its left and right sides pinched to single vertices(solid lines), and the rectangle Rh with its top and bottom sides pinched(dashed lines) With additional edges f f' as shown, the solid and dashedplane graphs are chinl to each other

Figure 3 shows that Lemma 1 is a special case of the following resultfor a general plane graph and its dual

Lemma 2. Let CI be a graph drawn in the plane, and let GI * be its dual,with edge set {e* : e E E(G)} Let f be any edge of G, and suppose thateach edge e 7,4 f of G' s assigned a state, open or closed Taking an. edgee* to be open. if e is closed and vice yen sa, Mlle, there is a path in C;consisting of open. edges and 'joining the endpoints of f on there is a path

56 1301h1percolotani Oil Ea the Hari is- Kasten Theorem

in G" — consisting of open (dent) CINGS and joining the endpoints off* Paths of both types cannot (I:tit:I shin/Una cously

Proof The proof is the same as that of Lemma 1: replace each vertex

of G with degree et 1w a Hack (topological) 2d-gon, and each degree

Mertes of G* by a white 2d-goo Replace each pair le, el of dual

edges with a 4-gon, shining edges with the polygons corresponding to

the endpoints of e and c*, respecting the cyclic order of the edges arid

faces around each vertex of Gi and Cr, as in Figure -1 Colour all 4-

Figure -I A plane graph C (solid circles and lines) di awn with its dual CI(hollow circles and clashed lines) There is a black 2d-gon for each degree dvertex of C. and a white 2d-gon for each degree d vertex of a' There is a-I-gon km each edge/chu rl-edge pair {c. this is black or white according towhether c is open or closed, apart from the hatched 4-gon, corresponding tothe distinguished edge f and its dual f•

goes except, that colitis/muffing to { il"} black or white irr ai ty way

Then in the inter face graph hunted by the sides of polygons separating

a black region horn a white one, ever y vertex has degree 2 except the

font vertices of tine -4igont coliesponding to f. f*}: the rest of the 1)1001

is as before q

cm fl ing to the probabilit y meastue = FPI! , p in which each bond of

is open with in obabilit y p independently of t he other bonds, Lemma 1

has the follow ing immediate consequence

8 I The Russo-Segnomm Welsh method 57

Corollary 3. 0) If I? and R' are k by / - 1 UM/ k - 1 by C rectangles

in Z2 . respectively, then

PtaH(R)) -F E I-v( / 01 1) = I

Ii is an It + I by II rectangle then

P t12( H ( R )) =112 (1)

(iii) If 8 is an n by it square. then

[2( 1? (S)) =P tp2( H ( S )) � 112

Pi vot' For part (i), recall that each bond of A = Z 2 is open indepe n-

dently with probability p, and the dual e' of c is open if and only ifis closed By Lemma 1, every configuration w lies in exactly one of

H(R) and V(R h ), so Pp (H(R))+ Illip (V(Rh )) = 1 But Rh is a k - I byt] rectangle in A. where bonds are open independently with probability1 - p, so Pp(17(/?h)) =

Ruts (ii) and (iii) follow immediately hour peat (i) q

It is easy to deceive oneself into thinking that (1) shows that par = 1/2Although self-duality is of course the reason -why' Pu = 1/2, a rigorousdeduction is far from easy, and took twenty years to accomplish

3.1 The Russo-Seymour--Welsh method

The next ingredient we shall need in our proof of the Harris-KestenTheorem is some toms of the Russo-Seymour Welsh Theorem relatingcrossings of squares to crossings of rectangles The proof we present ishorn Bollobtis and Riordan [2006c]

Lemma 4. Let I? = x [2,11 nn > n be an in by 2n rectangle. Let

X(R) be the event that them are paths P 1 and Pt of open bonds, such

that Pi CIOSSCS then by n square S = En] x (Id from top to bottom, and

P, tics within I? and loins some site on P 1 to some site on the light-hand

side of 2 Then IP1,(X(R))> Pp(H(R))11",,(1-(5))/2

Proof Suppose that V(S) holds, so there is a path Pr of open bondscrossing S horn top to bottom Note that any such Po separates 5 intotwo pieces, one to the left of Po and one to the right Let LI: (S) be theleft-most open vertical crossing, when one exists.. defined analogoushto the top-most open horizontal crossing discussed in the remark after

58 Bond pemolation OP V - the Hartis--Kesten Theorem

Lemma 1 By that remark, for any possible value P I of L17 (.9), the event

{LII(S) = } does not depend on the states of bonds of S to the right

of PI

We claim that, for any possible value Pi of EV (8), we have

Pp (X(R)ILV(S)= PO Pp(H(R))/2

To see this, let P be the (not necessarily open) path formed by the

union of 13 and its reflection PI' in ti re horizontal symmetry axis of .R,

with one additional bond joining P1 to P(; see Figure 5 This path

Figure 5 A rectangle 1? a al square S inside it, drawn with paths (solid curves)whose presence as open laths would imply X(R). The path P formed by Piits reflection P;, and th r single bond joining then, crosses 1? from top to

trot tom

crosses [Id x [il] from top to bottom With (unconditional) probability

Pp (11(R)) there is a path Pa of open bonds crossing 17 from right to left

this path must meet P By symmetry, the (unconditional) probability

that some such path first meets P at a site of PI is at least Pp(11(R))12.

Hence the event Y "(PI ) that there is an open path P in 1? to the right

of P joining some site on P i to the right-hand side of 1?- has probability

at least Pp (H(R))/2 But Y(P1) depends only on the states of bonds to

the right of P All such bonds in S are to the right of P1 in 8 As the

states of these bonds ale independent of {L1 7 (8) = P1 1, we have

Pp(Y(Pi) I LV(S) = P1) = Pu(Y(P1)) Pp(H(R))/2

If I t (PI ) holds and LI/ (S) = Ph then X(R) holds Thus

IFF,(X(/?) I Lli(S) = IP,„(H(R))/ 2

As the event V (S) is a disjoint union of events of the form {L1 7 (S) =

3.1 The Russa-SeyMaill -IVelsh Method 59

we thus have P (X (R) I V(8)) > INH(R))/2, and the result follows

Lemma 4 allows us to bound from below the crossing probability ofsome non-square rectangle; in particular, of a 3n by 2,1 rectangle. Leth i,(rn,n) = (11(R)), where R is any in by n rectangle in Z 2 and leth(tn,n) = h 1/2 (711, n)

Corollary 5. Fat all n > 1 we have 143, 1 ,20> 2-7

li

Figure 6 'if -Wu 2„ by 2n (square) rectangles R, and an a by 11 square 5 intheir intersection Zile solid lines indicate paths witnessing the event A(/?),and the corresponding reflected event for R.'

Proof Consider two 2n by 2n (situate) rectangles R' arranged as inFigure 6, and then by n square S in their intersection. Let X'(R!) be theevent defined analogously to X(R) but reflected horizontally ApplyingLeunna 4 to the rectangle 1? (which happens to be square),

PiLX (Jr )) = P„ (X (T)) Pi,(11(R))Pp(1/(S))12

The events X (R), X' (R') and H(S) ate increasing, and hence positivelycorrelated by Harris's Lemma (Lemma 3 of Chapter 2) If all threeevents hold, so does H (R U R') Thus,

h(3n, 2n)

P1/2 (H(R U R'))

• 1111/2(X'U1'))P112(X(R))P112(H(S))

• Prp2(H(R))2P1/2(1/(S))2P1/2(H(S))/4.

But 1? and S are squares, so. by Corollary 3.

h(3n,211) (1/2) 2 (1/2) 2 (1 /2)/4 = 2 -7 q

60 Bond percolation an E2 the Han hi - Kasten Theorem

It turns out: that the difficult step is getting horn scrim/es to elongatedrectangles: ft QM COI ollm V 5 it is vent' easy to deduce Hatt is's Theorem

Indeed, considering in ) by 2n and mo by 2n rectangles intersecting in a

2n by 2a square as in Figure 7, by Harris's LCIUMI/1 we have

Figure 7 Two rectangles intersecting in a square II the rectangles have openhorizontal crossings, and the square an open vertical one, then the union of

the iect angles has an open horizontal crossing.

h(m )- 2)1.2n) > h(a 2n)h(ne).2012 (2)

Ion ill > 2n hi particular,

Nat 20> h(a).2n)l)(3m2t))/2 > 2-sh(m.2n),

so h(lta. 2a) > 2 17-8k lot all k > :3 and n > 1 As h(m.2n 1) >

h(n).2n) it Follows that thew are constants h k > 0 such that

li(km a): Il k(3)

for all k > 2 and a > 1Alternativeh, starting with an nt i by 2n rectangle R and tun no, b n

2ti rectangle the proof of Lemma 1 actually shows that

Mtn ) T ne)- a. 2n) > h(m).201)(m)). 20/25

lot in tn, > 2u Plats

h(511.20> h(311.2n) 2 / 9r1 > 9-19

rind

1461).20 > 11(511.2014 9 m 90/95 > 9-19-1-5 = 0 - (1)

As 2- 1 < /1(3n. 2n) < 1/2 lot ever y n. it is natural to expect that

1 1 (31 1 .2a) coin el ges to a limit as n — More gemnalh, one would

expect Nun. fro) = (a lb) lot sonic function /(:c) with 0 <

f(r) < 1 Indeed. it would be astonishing if this were not the caseSurprisingly this conjecture is still open In fact. this is a very special

Hurt is 's Theorem 61

case of the general conformal invariance conjecture of Aizemnan andLanglands. Pouliot and Saint-Aubin f10941 This conjecture was proved

Surnnov (2001a1 for site percolation in the triangular lattice; seeChapter 7

3.2 Harris's Theorem

From crossings of long, thin rectangles to Harris's Theorem is a very

short step Let us wr ite r(Co) for the radius of the open cluster contain-ing the origin, so

(C0) = sup{ d(ri. (I) r E

where d(x. y) denotes the graph distance between two vertices of Z2

Theorem 6. Fm bond ',emulation in V 0(1/2) = 0

Moot' In fact. we shall prove slightly !now. : namely that

P i/2 (1 (Cid fri (5)

fin all n > 1. : where c > 0 is a constantAt, p = 1/2„ the bonds of the dual lattice A" are open independently

with probability 1/2.: so. hour (4), the probability that a tin by 2r i rect-angle in A' has an open crossing is at least 2- =5 Consider two 6[1

by 211 and two 21i by 6n rectangles in A-, arranged to form a 'squareannulus as in Figure 8 By Harris's Lemma, with probability at least

Figure S Four reel angles Fuming a square annulus

.r• = 2 -i00 > 0 each rectangle is crossed the long way by an open (dual)path Il this happens, then the union of these paths contains an opendual cycle surrounding the centre of the annulus; see Figure 8.

For > 1. let A t; be the square annulus centred on (I/2,1/2) with

G2 Bond percolation on 7S 2 - the Harris-K oleo Theorem

inner and outer radii 4 k and 3 x 4 k , and let Ea he the event that Ak

contains air open dual cycle surrounding the interior of A k , and hence the

origin. Then P(Ek) > e for every P. As the A k are disjoint, the events Ea

me independent If Ea holds, then no point inside .4 k can be joined toa point outside A k by an open path in V. so r (Co) < 1 +3 x 4 k < 4k+ I

Thus,

P I/2 ( I 4c±1) P1/2 n E = H P112( Ed � 0\ k= k=

and (5) follows Also, for any

0(1/2), IP 1/2 (I (Co) = x•)) 5- Pr/_ (r (Co) n-c

so 0(1/2) = 0 0

Let S he an n by n square If H(S) holds, then at least one of the a

sites v on the left of S is joined by an open path to a site at graph distanceat least rr from r' It follows that Pr/.,(H(S)) < tiP i r)(1(C0 ) >

Hence, It/" e (Co) ) > a) > 1/(2n) for all a > 1. It is natural to expect

that P I 1 .)(1 (Co) > decays as a power of n, i e., that the limit

log P 1/2 (I (C0) � (1) lim

log n

exists Once again, this natural conjecture is still open, although a cor-responding result for site per colation on the triangular lattice is known:see Chapter 7 The limit above, if it does exist, is often denoted by 11p,

or 1/(5,. It is one of the critical rap(111111115 associated to percolation; seeChapter 7.

Looking back, the most difficult step in the proof of Theorem (i isLemma 4 or the equivalent results of Russo [1978] and Seymour andWelsh (19781 Although Harris's proof is very different from that pre-sented here, a key step is similar to a step in the proof of Lemma 4:roughly speaking, given an 'outermost open semi-circle 8 around theorigin' within a certain region, one reflects this to form a cycle S U .Then a path meeting this cycle from inside is as likely to meet the 'real'part .5' (all of whose bonds are open) as the reflected part S'

It is pr rssible to prove Hall is's Theorem using Lemma 1, Harris'sLemma, and the independence of disjoint regions, without ever consider-ing left-most crossings or then equivalent; such a proof has applicationsin other contexts, where the proof of Lemma 4 does not work Indeed, as

we shall see in Chapter 8, a (long) proof of this kind was given for random

3 A sharp transition 63

\Tor onoi percolation by Bollohas and Riordan (2006a] In that setting, noresult: directly equivalent to Lemma 4 or to the Russo-Seymour-WelshTheorem is known

3.3 A sharp transition

Fahly ear ly on it was recognized that there is an absolute constant e < 1for which the following statement easily implies Kesten's Them ern: forany p > 1/2, there is an T1 = a(p) such that Pp (H(R))> c for a 211 by nrectangle R. For example, Chaves and Chaves 11986b1 gave an argumentshowing that c = 0.921 will do The factor 2 here is not important.:recalling that h p (nt,n) is the Pp-probability that an or by n rectanglehas an open horizontal crossing, it follows from Harris's Lemma that ifh 1 ,((1 + 5)n, 1 as n a for some E > 0. then h p (Clf n) 1 asn for every C > 0.

We shall give ar t explicit lower bound on h p (tma), using a sharp-threshold result of Ft iedgut and Kalai, Theorem 12 of Chapter 2, and theAlmgulls-Russo formula, Lemma 9 of Chapter 2.. We start by boundingthe influence of a bond in a rectangle I? on the event II(R)

In the context of percolation, a bond e is pivotal for an event E in aconfiguration w if precisely one of w and ur is in E, where ,./r± are theconfigurations that agree with w on all bonds other than e, with e openin to + trod closed in co- . In other words, e is pivotal if changing the stateof e changes whether F holds or not The influence of e on E is

Ip (e,E) = Pp ( e is pivotal for E )

If E is increasing, then

Ip (e,E) = (w 6 E. co--

Lenona 7. Let R be an rn by a rectangle in 27.? and let e be a bond in

I? Then.

11,(e, MR)) 2P 1/2 ( r(Co) _� 4 1 1 1 12 , (n

)/21) (6)

for all 0 < p < 1

Proof Throughout the proof we work entirely within R. consideringonly the states of bonds e in I? Suppose that, a bond e in I? is pivotalfor the increasing event 11(I?). As LO + E II(R), in tire configuration (d4

there is am open horizontal crossing of R Since ur 0 1-1(R), any such

Band per rotation on the Hal is - esten Thew um

mussing must use c Hence. in the configuration w one endpoint of isjoined I A an open path to the left of R. and the (Whet to Hie tight: seeFigine 9 Thus at least one endpoint of e is the shut of an open path

igwe 9 A bond e (joining the solid discs) that is pivotal lilt 11(1?): the dualbond c' (endpoints shown as ( l osses) is pivotal for Die)

of length at least 10 /2 – L so

0 (e 11(11)) < 2P(1(C0 ) > 111/2 – I) (7)

As Lt.– ii-I 11(R). IA Lemma 1 we have Lc – E V(Rh ) Siinilm iv.

1 ,), so in y t lime is au open dual path ()tossing Rh vertically andusing the edge e* dual to e Hence, in it). one endpoint of c' is the :Wm tof an open dual pat h of length at least (a – 1)/2 Since dual edges ateopen with pi obabilih 1 – p. it Follows that

11(1?)) < (CO>(rt– 1)f2)

(8)

Fin ant a the event 1(C0 ) > a is incteasing. so P p () (Co) > a) isan inmeasing function of p Thus (6) Follows immediately bou t (7). tot

< 1/2, and flow (8). p > 1/2

Lemma 8. Let p > 1/2 and an allege, p > 1 be 1i.:ecd There artconstants 7 = 7 (p) > 0 and 0 0 = 0 0 (p, p) such that

11 0 (pir 1 )> 1 – 11 – "' (9)

fat all 0 > au

Proof Let 1? 1) 0 a pa ht n iectangle nom (3). we have

Pt/2(1-1(R)) h (10)

3.3 .1 slung transition 65

ha some constant > 0 depending only on p Rom Lemma 7 awl (5).

for > 2 we have

1„e(e,1-I(R))< =

for ever y bond e of I? and all p E [1/2,p], where a > 0 is an absoluteconstant Writing f (pi ) for Pp,(11(R)), flour the Friedgut-Kalai resultTheorem 12 of Chapter 2 it follows that

( e , H ( R )) (100(1 — (0) log(1/6)tell( TO

for all p' E [1/2, p]. where e > 0 is an absolute constantBy the Margolis lbws° formula (Lemma 9 of Chapter 2). the stun

above is exactly the derivative of I (p') with respect to p' 'Thus, writing

g ( p9 lei log ( V)/( 1 — 1(0)),

- 100> clog( 'WO) = en: log a..di/ 1(7)(1 — (0) "

horn (10), g(1/2) is bounded below by ir constant that depends onp Hence. for rt > ag(p), we have g(p) > ac(p - 1/2) log n, say awl (9)follows q

Using Iris weaker 'approximate 0-1 law ! instead of the mom iecientFriedgut Kalai result. [1982] proved a weaker form of Lemma 8:this weak form is more than enough to deduce Heston s Theorem

Following Bollobas and Riordan (2006ci. we give an alternative proofof Lemma 8, using a version of the friedgut- Falai result for symmetricevents.. Theorem l3 of Chapter 2 The other ingredients of this proof areHarris's Lemma and (3) tins. rather than (5) The idea is that. if II(1?)

were a s t innwtric event. then Lemma S would follow immediately from(3) and the Fr iedgut-lialai result Of course, 11(1?) is not symmetric,but it is very eas y to convert it into a suitable symmetric went

Alternative proof of tetanal 8. Fix p > 1/2 Rom (3), there is anabsolute constant 0 < < 1/2 such that P i p,(11(R)) > c for any -fnby rt rectangle angle I?

For n > 3. let T2, be tire graph C'„ x obtained from Z2 1w iden-tifying all pairs of vertices for which the corresponding coordinates arecongruent, modulo This graph is often known as then by a discrete

torus Note that has 112 vertices and 2n 2 edgesFor 1 < t < n-2, a Ai by I rectangle 1? in is all induced subgraph

of corresponding to a k by I rectangle 1?' = [0+1, ads x [5+1. b+11

0

1=1

66 Band percolation on Z2 - the Ilan is- Kesten Theorem

in 22 Out rectangles in the torus are always too small to 'wrap ill Olin( II,so, as in the plane, any such subgraph is an abstract k by t! rectangle.

We shall work in T2 = 'ffiL, taking the bonds to be open indepen-dently with probability p. We write PI, Pi 2 for the correspondingprobability measure

Let E„ be the event that: 1 2 ti„ contains sonic 4o by rectanglewith an open horizontal crossing, or some T/ by 1n rectangle with an openvertical crossing Then E„ is symmetric as a subset of 1)(X), where Xis the set of all 50n 2 edges of

Considering one fixed -In by o rectangle Pr in 1 2 , we have

P I: 22 ( EO ?_Pi;2(H(E)) = P o2( E(E)) > (.12

Let /5 = (p - 1/2)/(25c,), where c 1 is the constant in -Theorem 13 ofChapter 2, and set E = I/-" (16 As (5 depends only on p„ there is anno = ro ) (p) such that E < e2 < 1/2 lb/ all a > no Now

log(1/5-) log(1/(2E)) —9 = 25c,(5= „ >

log(n-) log(50n2)

Hence. by Theorem 13 of Chapter 2 . as P ,(E„ ) > > E we have

Pil-/(E„) > I - 2 = 1 - /1-50 (11)

for all n >Let R I Mr, be the 3 1 I by 2/r rectangles in 1- whose bottom-left

coordinates are all possible multiples of 11 Then any 4o by a rectangleR. iu 12 crosses one of the Er in the sense that the intersection of 1?and .R.; is a 3n by ri subrectartgle! of R.; Similarly, there are 211 by 3nrectangles 000, R50 so that any n by 41/./ rectangle in 12 crosses oneof these hen/ top to bottom

It f loflovks that if E„ holds, then so does one of the events E„ ,i , i =50, that Ri is crossed the long way by an open path Thus E;;..

tire complement of E. contains the intersection of the El;Applying Harris's Lerruna (Lemma 3 of Chapter . 2) to the product

measure PI' , For each i the decreasing events E,t, and r) Et are

rP

Thus, from (11), for n > no we have

Pi; ( Eli < /I/5(

Kesten's Theorem 67

so Pi; (E„ ) > 1 —Now E„ is the event that there is an open horizontal crossing of a

fixed 3n by 2n rectangle R in the torus. which we may identify with acorresponding rectangle in V Thus

P„(H(R)) = Fp(H ( R)) > 1 —

whenever R. is a 3n by 2n rectangle in 27,2 and a is large enough. 'Using(2), it is easy to deduce (9) q

The proofs above involved various explicit bounds. These ate notreally relevant As noted earlier, the following much weaker form ofLenrnra S is enough to deduce Kestert's Theorem.

Lemma 9. Let p > 1/2 be fixed If 1?„ is a 31 1 by n rectangle to V,

their P„(II(R„)) 1 as n pc,

This may be proved by either of the methods above Indeed, it followsfrom Harris's 'Theorem that the influences of the bonds e on the event11(R) tend uniformly to zero as the shorter side of R tends to infinity.Using a qualitative form of Theorem 12 of Chapter 2 (that the stunof the influences is at least 1(S) if all ale at most O. with (6)

as t (I), or Russo's approximate 0-1 law, it follows that the stun ofthe influences tends to infinity Using inf„ h lt )(3n, > 0 (from (3)),Lemma 9 follows Alternatively, one may use (3) and a qualitative for inof the symmetric Friedgut -Kalai result, working in the torus as aboveEither argument shows that, unlike Lemma 8, Lemma 9 does not dependon the particular form of the Fr iedgrit-Kalai bound.

34 Kesten's Theorem

As noted earlier, it is well known that Kesten's Theorem follows eas-ily from Lemma 9; inn fact, from the quantitative form of this lemma,Lemma 8, Kesten's Theorem is more or less immediate

Let E,— denote the event that there is an infinite open cluster Recallthat P„(E„ ) > 0 implies that = 1 and 6(p) > 0

Theorem 10. Fat hand percolation in V. p > 1/2 then 1P0 ( Etc ) =

Proof Fix p > 1/2, and let, 7 = -l (p) and no = no(p,2) be as inLemma 8 Let > P.0 be an integer to be chosen below. For k =0,1,2, , let Rk be a rectangle whose bottom-left corner is at the origin,

68 Bond pnroloboa on V the Han Pe-AR:sten Theorem

with side-lengths 2 /1 11 and 2 k -11 n. whew the longer side is vet ical if k is

even and hot izantal if k is odd: see Figure 10.

Figure In 11- he mciangles En to / (00 not labelled) drama o f en pathscut responding to Lhe OVentS Er;

Let .E/ be the event, that ./? k is crossed the long way by an open path

Note that it'll two such ctossings of Ilk and RAH' must meet, so if all

the Eh. hold then so does Ex . If it is huge enough then, by Lemma 8.

= < 1,— 9

6>a k

so P ,(Ex. ) > F,An k ,.„ > 0

Together Fhetnems 6 and 10 show that //n(Z 2 ) = 1/2

Staffing nom the \wake( Lemma 9, a little mo l e work is needed

Again (hoe ale several possible arguments One is a 9enormalization'

tugument due to Aizenman, Chaves, Chayes, Frohlich and Russo [1984

see also Chaves and Climes [19861)]

Proof of Theorem 10 strum/ vet (inn, Fix p > 1/2, and recall that

h p (m. n) denotes the Pr,—probability that MI m by a rectangle in :12

has an open Inn izootal classing Consider three 2n by n rectangles

aye/lapping in two it by n squares as in Figure 11 Noting that the

jnobabiliN that nun In a squaw has an open vertical classing is just

h ijn nom Ilatris's Lemma we lnae

l /.(-Lm > 031),,(n. n) 2 > h p (211.0 5 (12)

Placing No disjoint in b y n rectangles side by side to form it -In by

liesten s Thcomn 69

Figure Four 0 by rr squares in 2 2 Each consecutive pail fin ins a 2a byrectangle It these three rectangles have open horizontal crossings, and the

middle two squares have open vertical crossing,, then the union of all Emusquares has au open izoutal mussing

ELIM11111411

1 'gum 12 ly lis by n formicu e I iv 2n rectangle

2 p rectangle as in Fig u re 12 t he elect s that eaclr has an open hotizoul al

mussing ale independent Hence,

h p (-1n, 2 0 ) > 1 - (1 - ›i_(1_hp(2„,„),)- (13)

Writing h p (20, n) as 1 - 5. from (13) we lime h p (-1n, 2n) > I - 2552,which is at least I - ,F/2 if 5 < 1/50 By Lemma 9 there is an nwith /i,,(2 p , > o) > 0.98: it hallows that hp(2kijii n,2/'M) >- 2 -k/50 tot all k > 0. and liPi,(E) > 0 Wilms as in the first proof of

'Theorem 10 q

The argument above shows that if for a single value of n we have

0) > 0 951 (a loot of :r = 1 - (1 - ii 9 ) 2 ), then 0(p) > 0.In fact, arguing as in Chaves and Chaves 11.9866], one eau do a little

better Dividing a 2n by it rectangle into two ve/ t es-disjoint squates, ifthe rectangle has an open horizontal flossing, so do both squares As thesquares ale disjoint, the events that they have open hot izmital crossingsate independent,. so h i,(2m n) < h p (n, n) 2 Therefore the first inequalityin (12) implies that h p (-bb > 0)1, and the first inequality in

Bond percolation on. Z2 the Harris-Kesten Theorem

(13) gives h p (tn.2n) > 1 – (1 – /1 11 (2m0-1 ) 2 Thus the value 0.951may be replaced by 0.920 , a root of x = 1– (1 –1, 1 )2

3.5 Dependent percolation and exponential decay

In the final section of this chapter we shall prove that for any p < 1/2 we

have P1,(1(0 1 > n) < exp(--an) for some a = a(p)> 0 This result is dueto Kesten [1980; 1981], although, as we shall see later, it inflows easilyfront Kesten's result that p! (Z2 ) = 1/2 by an argument of Hammersley[195Th] Once again, (Mete are several possible proofs; we shall use theconcept of dependent pet colation -- this will be usellil in later chaptersas well

Let G be a graph, and let 115 be a site percolation measure on G. i e aprobability measure on the set of assignments of states (open or closed)to the VCI flees of G The 11leatill/ e P is k-independent if, whenever 8 and

T are sets of vertices of separated by a graph distance of at least le, thestates of the vertices in ,5 ate independent of the states of the verticesin T 'The tenn k-dependent is also used by some authors For k 1,the condition is exact l y that the states of the vertices ate independent,so the first non-trivial case is k = 2

Liggett, Schorunann and Stacey [1997] proved a general testa com-paring k-independent measures with product measures We shall needtwo simple consequences, which ate essentially trivial to prove (Meetly.Recall that, when considering site percolation on a gtaph G, we write C„

tot the open cluster containing e, i e , ha the set of sites of CI connectedto !' by paths all of whose vertices are open

Lemma 11. Let. k > 2 and A > 2 be positive integers. and let G he

a (finite a t graph wtih matintarn degree at most A There are

constants P i = pr (k. A) and n = a (k. A) such that if P is a k-independent

site percolation ineas itir all G in which each site is open with probability

at MOW then

11; exp(–an)

for all vet trees v of C and all n > 1.

FM tit er more, fa; It t (2) we may take any constant such that the quan-

tity eil(p i /(41+1 ) is less than 1

Proof If IC, > n. then the submaph of 6' induced by the open verticescontains a tree T with n vett ices. one of which is ti It is easy to check

3 5 Dependent percolation and exponential decay 71

that the number of such trees in G is at most (eA)"- 1 ; see Problem 45 inBollobris [2006] Fix such a tree T If w is any vertex of U. then at most

1 + A + + < A k vertices of G are within graph distance k -1of to. Hence, there is a subset S of at least n/A k vertices of T such thatany E S are at graph distance at least I-; indeed, one can find sucha set by a greedy algorithm, choosing vertices one by one The verticesof S are open independently, so the probability that every vertex of Tis open is at most 14'91 . Hence,

p;'/ < (cAp) 7/

I/AkChoosing pm small enough that = GAT), < 1, the resit!t, follows,with a = - log;.

For the second statement, note the quantity A k above is in fact an

upper bound for 1 -f- A + • + At I Hence, when /c = 2, it may bereplaced by A + 1 fj.]

In the proof above, we could have assumed without loss of generalitythat k = 2, replacing the graph G by its (k - pawl, i e., the graph

(7 (k -' ) on 17(G) in which two vet tices are adjacent if their graph distancein C; is at most k - 1

Using Lemma 11, one can easily deduce Reston's exponential decayresult for bond percolation on 1 2 from Lemma 9. The basic idea is toassociate a large square 5, to each v E 12 , and to assign a state to 0depending on the states of the bonds in (and near) S,,, in such a waythat the states of sites v separated by at least a certain constant distancek ate independent, each v is unlikely to be open. and percolation in thebond model implies an infinite connected set of open sites

Theorem 12. In bond percolation on 12 , fat evecy p < 1/2 there is a

constant a = o(p) > 0 suck that IP0 (]C0 1 > JO< exp(-an) for all n > 0

Proof Fix p < 1/2, let pr = (5,4) > 0 be a constant for whichLemma 11 holds with k = 5, A = 4, and set c = (1 - pl)1/1.

Let A = 12 and let A" be its dual, so the bonds of A* me openindependently with probability 1-p As 1- p > 1/2, by Lemma 9 there

is an in such that hm_ p (3111,1n) > c, so the probability that a 3m byor rectangle in A'' has an open horizontal crossing (of dual edges) is atleast c

Set s = nr +1, and let 5' be an s by s square in 72. We can arrangefour 3t11 by or rectangles in A" overlapping in in by ta squares, so that

NA] < (eA)

72 Bond percolation on 2: 2 - the Halt issICeslen Theorem

then onion is an annulus as in Figure 8 The inter for of A containssquare in A with in + 1 = s sites 00 a side, which we shall take to be ,913v Harris's Lemma, t he probability that each of the Wm leo angles iscrossed the long way by a path of open dual edges is at least e t = 1 —puThe union of font such ctossings contains an open dual cycle sin !omittingS. Let B(S) he the event that some site in S is connected by au openpath to a site at L, distance s horn 5' Such an open path in A cannot.(loss an open dual cycle in .1 surrounding .S. so we have Lau,(/3(S)) <

Informall y, we define a site pet eolatitm measure F on Z 2 k: takingeach v = (di it ) E Z2 to be open if and only if B(S) holds fin the ss squats = 1st' + 1. sr x Is// + sit + Mot e for walk, let Al

denote the independent bowl percolation !node! on Z- in which emitbowl is open with probability p \\e define a site petcolatam modelAl on 222 as follows Let j: 2 E1(7:2) — 2" :1211 be the function [tom thestate space of A/ to the state space of Al that we have just defined. so

(w))(v) = 1 if and out il, in the configtuation w. the event B(8, )holds The function , J" and the measure 111 1, induce a probability measure111 on 2 1 CL1 ' given by 1111 (A) = [ —I LA)) This measme on 2 1 C1:2)a . gives us a site percolation model A/ on 2 1 I:1-)

Since the event B(S,.) depends out on the states of bonds within1„-distance s of So .. Hirt/treasure F is 5-independent: nu thei moleeach eE.= is open with P-plobabilit n Pi,(B(5,)) < g, Let co be theopen cluster of the origin in out original howl pet colation Al. and let (17,he the open cluster of the origin in the 5-independent site percolationmodel Al II n Lemma 11 thew is i1/1 a > (I such that

<

of eve/. vIf 1C1/d > Is + 1) 2 . then el. el \ site n o 1 Co is joined by au open path

to some site at 1.,-distance 2s flow w If in E 8, then it follows thatB(.9,.) holds Tints, if rid > (-1s+ 1) 2 . t hen B(S,,) holds tot every c suchthat contains sites of Co 1 he const uction of the model Il go es usa mutual coupling of Al and Al: a site c is open in A/ if and outs ifS(B,.) holds ill Al. III particular. event ' with n FL- 0 is open in Al.

so the set of such v forms an open cluster iu Al. and is thus a subset 01Co Hence. as each So contains onl y s 2 sites. fi g n > (-Is + 1) 2 we have

F,,000 g ) 111 0(.7,1 g /s2 ) exit( --ust/s2).

completing the woof of Thement 12

d 5 Dependent percolation and exponential decay

Theorem 12 immediatel y implies that \ (p) cc... for p < 1/2. so> 1/2 Together with Theorem 19. this implies tire Harris- Kesten

Theorem

Theorem 13. 14 (22 ) = pli I/ (Z2 ) = 1/2 q

Later. we shall see that pr = m i holds in a very general context In fact.Merishikov [1986] proved exponential decay of the radius (and, under anadditional assumption. of the volume) of Co for p < p H . again in a verygenet al context

117e finish this chapter by noting that dependent percolation gives yetanother way of deducing Nesten's :(irearm/1 hour Lemma 9: again, theIces lemma will be useful in other contexts. This time we consider bondpercolation A bond percolation measure on a graph Cis li-independent

if the states of sets S and T of bonds are independent, WIRMICVC/ .9 andT. ate at graph distance at least A . This time, the case k = 1 is alreadynon-trivial. Indeed. E01 A! 1 the separation condition is exactly (hatno bond in S shares a site with a bond in

Lemma 14. There is a po < 1 such that if is a I-independent bond

percolation ni ewiare on 23 2 in which each bond is open with probability at

least po then 1111(70 1= ) > 0

Proof Suppose that the open cluster Co containing the origin is finiteThen by Lemma 1 of Chaplet 1 there is an open dual cycle 8 1 sin I oursl-ing the origin, of length 21, say As shown in tire proof of Lemma 2 ofChaplet there are at most

(po i _ I < —9

3-

dual cycles of length 21: sou rounding fl. where I r k < -1 x 3 k - I is the numberof paths of length lt ill 22 starting at 0 Let S denote the set of duals ofthe bonds in 5', so S is a set of bonds of A = Z 2 As the graph 22 is1-edge colourable. there is a set of at least 1/2 Vertex-disjoint bonds hr

diet-I.-probability that S' is open. i e the II-probability thatall bonds in S me closed, is at most (1 - po)112.

It Follows that the probability that Co is finite is at most the expectednumber of open dual cycles surrounding the origin. which is at most

v 3210 - pot129

t>2

7-1

Band percolation on V the Harris-Kesten Theorem

If po is close enough to 1 (for example. po = 0 997), then this sum is lessthan 1 q

As stated, Lemma 11 is essentially trivial; it is also immediate fromthe general comparison result of Liggett, Schonmann and Stacey [1997].In applications, the value of po is frequently impor taut Currently, thebest known bound is given by the following result of Baster, Bollobasand Walters [2004

Lemma 15. If Pis a 1-independent bond percolation Incas 1171! on Z2 inwhich each bond i 9 open with probability at least 0.8639, then Pa(,'ol:Tic) > 0 q

Bollobas and Riordan [2006b] pointed out that Lemma 11 may beused to give yet another proof of Kesten's Theorem; lot this, the valueof po is irrelevant.

Proof of Theorem 10 third version. Let p > 1/2 be fixed, let. po < 1 li)ea constant for which Lemma 11 holds, and set c p(1/ 3 Given a 3n bya rectangle H. let S' and S" be the two end squares when 1? is cut intothree squares. Note that 11(1?) certainly implies II(S' ) so, by Lemma 9.

Pp(1/ (S")) = = P0 ( 11 ( Si )) Pp(H(R))

if n is large enough. which we shall assume born now onLet 0(1?) be the mem H(1?) f1 fl V(.9"): see Figure 13 By

aFigure la A au by n rectarectangle It such that il) holds.

Harris's Lemma.

Pri (G(R)) > P„(//(R))ifii,(17(51))Pp(V(S")) c3 = po

Define 0(1e) similarly for an n by :31/ rectangle, so that by symmetrywe have Pp (C(R')) = Pv(0(1)) po

Writing Al for the bond percolation model in which each bond ofV is open independently with probability p, let us define a new bond

3 5 Dependent percolation and exponential decay 75

percolation model Al on V as follows: the edge from (x,//) to (x +1, y)is open in Al if and only if C(R) holds in Al for the 3n by n rectanglePrix +1,21/x-1-3n] x [2ny + 1, 2ny + id Similar ly, the edge from (.e, y)to (:e,p 1) is open in Al if and only if G(D') holds in Al for the n by311 rectangle [2nx + 1, 2rix + x [2ny 1, 2ny

Figure IA A set of open edges in A/ (left) Hid

responding rectangles R.drawn with C(1?) holding in Al

Let F be the probability measure on 21(:) associated to AI •Phenis indeed 1-independent, as C(R) depends only on the states of edges inR, and vertex-disjoint edges of Z2 correspond to disjoint rectangles

By Lemma 14, Pl I Col = > 0. However, we have defined 0(11) insuch a way that an open path in Al guarantees a corresponding (muchlonger) open path in the original bond percolation AT, using the factthat horizontal arid vertical crossings of a square roust meet; see Fig-ure 14. Hence, 1111,(E,) > PaCor = ) > 0, completing the proof ofTheorem 10 q

The ungurnent above works with 2n by n rectangles; the only reasonfor using 3n by rt was to make the figure clearer Also, in addition to thelong crossings, it is enough to require a vertical crossing of the left-handend square of each rectangle I?, and a horizontal crossing of the bottomsquare of each R'. As hp (rx n) > 11,,(2t, n) 1/2 , to prove percolation itthus suffices to find an n with h p (2n,n) 3/2 > Po, where pa is a constantfor which Lemma 14 holds Using the wane go = 0 8639 from Balister,BollobAs and Walters [2005], li p (2Thn)>. 0 907 . will do.

'16 Bond percolabon an 22 thc. Harris-lticsten Theorem

3.6 Sub-exponential decay

It is not hard to show that. for p > 1/2, with probability 1 there is aunique infinite open cluster; we shall present a very general result, of thistype in Chapter 5. fu this range, all other open clusters are 'small' withprobability close to 1; more precisely, the probability that n < < ccdecays rapidly as n increases. In analogy with the situation for therandom graph C7(n.p), cure might expect that this decay minors that ofopen clusters below the critical probability, i e., that P p < [C10 1 < (x)is approximately exp(—ai n) for some a' > 0 This turns out not to bethe case We shall present only the vett' simplest: result in this direction;stronger more general results have been proved by Aizenman, Deleonand Souillatd [1980 -1 among others: see Cilium/eft (19991

Theorem 16. In bond percolation on V for raw p > 1/2 there arr

constants b = b(p)> 0 card = c(p) > 0 such that

exp(—bli7) < < S.: ex t)(--- e 00 (LI)

for all n 1

Moat Tire upper hound is essentially immediate horn Theorem 12Indeed, suppose that [Col =Ir. and let O'C la be the external boundarrof Co, as defined in Chaplin .1 By LC1111118 .1 of Chapter 1. O'C'n isa cycle in the dual lattice + (1/2. 1/2) containing Co in its interiorThus, the area enclosed b y O'Co is at least II, so its length is at least2 \,iti But: era s (dual) edge in O'C'ü is open As O sc Ca crosses thepositive x-axis at sm ile coordinate 1/2 < :r < n — 1/2, we have shownH u tt whenever = there is some dual site v = (T. —1/2), 1/2 <

< n — 1/2, such that the open dual duster containing r' contains atleast 207 sites. As dual bonds are open independently with probability1— p < 1/2. b y Theorem 12 the probability of the latter event is at most(II— 1) exp(-2aiii) for some a > 0 Thus

BIRO) ICH <

(or — 1) exp(-2a

which is at most CX1)(—blii) for all n tot some b > (IFM the lower bound. suppose that n = and let 5' be the I by C

square [0, f — x C — For each of the n sites ,r in 5', let: Es be theevent that there is an open path horn :r to some site in the boundaryof .5; see Figure 1.5. Note that I.L1 holds triviall y for all sites in theboundary of Whenever Cr is infinite, then holds, so I"„ (E,) >

36' Sub-6 nmenbal decal'77

Figure 15 A 9 by 9 square .5 with bottom-left coiner tire aright. drawn to-gether with all the open hands in its interior 'The filled circles are the sites .rfor which E„ i e.. those joined to DS by an open path. If the bonds in[IS are open, and the dual bonds surrounding .9 (dashed lines) ale also openthen the open cluster ()I' the (night consists precisely of time filled circles

0(p) > Let N be tire number of sites :It E ,5' for. which E . holds Then2 t (!fir) = 5— s E s Pn( E.,) > 110(p) As A < n always holds, it fellows that

PAN > n0(P)/2 ) 0(p)/2

Let F he time event that each of the -1(1 — I) boundary bonds of S isopen, and that each of the if bonds from .9 to its complement is closed,as in Figure 15. Then

Pn1 F ) = 04 — > (Phi — /d).'

Since each event Es depends only on the states of bonds in the interiorof S. the random variable N is independent of the event F. and

Pp (n0(p)12 roi <11c) 1?„(F n {N > nO(p)12})

Pp (11 )Pi,(1N nO(p)/20

(PG — (0)12

As f = this proves the towel bound in (14) for 11 a square Sincethere is a squats between n and 3 11 for all o > tire lower bound fru alln lid lows

In higher dimensions, i.e lot bond (or site) percolation in 3d t it isnot too hard to guess that Pp (n < ECM < cc) Nvill decay t oughly asexp(-10- 101 ), but this result is not so easy to prove; see Section 8 6of Gr itnuett (1999(

4

Exponential decay and critical probabilities- theorems of Menshikoy and Aizeninan

Barsky

Out ainr in this chapter is to show that, for a wide class of percolationmodels, when p < prr tie cluster size distribution has an exponentialtail.. Such a result certainly implies that pr = vu; in fact, it will turnout that exponential decay is relatively easy to prove when p < p r (atleast if -size' is taken to mean radius, rather than numberof sites) Thusthe tasks of proving exponential decay and of showing that: p = areclosely related.

Results of the latter t y pe were proved independently by Nlenshikov[1986] (see also Menshikev, Alolchanov and Sick)/ wilco [1986]) and byAizemnan and Barsk y [1981 under different assumptions Here we shallpresent Menshikov's ingenious argument in detail, and say only a re

words about the Aizennian-Bar sky approach. As we shall see, 1\ fen-shikoy 's proof makes essential use of the Margulis-Russo [Mrada andthe van den Bel g-ICesten inequality

4.1 The van den Berg-Kesten inequality and percolation

Let us briefly recall the van den Berg-Kesten inequality, Theorem 5of Chapter 2, which has a particularly attractive inter pretation in thecontext of percolation In the context of site (respectively bond) perco-lation, an increasing event E is one which is preserved by changing thestates of one or more sites (bonds) from closed to open, and a witnessfor an increasing event E is just a set If of open sites (bonds) such thatthe fact that all sites (bonds) in are open guarantees that E holdsFor example, considering hoed percolation on 52 , as in Chapter 3, let 1?be a. rectangle, and let 11(1?) be the increasing event that 1? is crossedhorizontally by an open path P Then a witness for . 11(R) is simply an

/ I The van den Bety-Resten inequality] and percolation 79

open path P crossing I? horizontally, or a set of open bonds containingsuch a path

The box product ED F of two increasing events is the event that thereare disjoint witnesses for E and F For example, 11(R) 0 H(R) is theevent that there are two edge-disjoint open paths crossing R horn leftto right. (For site percolation, the paths must be vertex-disjoint ) Aswe saw in Chapter 2, van den Berg and Kesten [1985] proved that forincreasing events in a product probability space, P(E 0 F) <P(E)LP(F)This inequality is the driving force behind most of the proofs in thischapter.

Let us illustrate this inequality with a simple application to bond per-colation in D i . Let p < hz = 4 (Z") be fixed: as usual, when the contextis clear, we do not indicate the specific model under consideration in ournotation Since x(p) =1E),(1C0 1) < there is an in such that

E„(ro n s„,000 1/2, (1)

where S„,(x) is the sphere of radius in with centre i e the set of sitesat graph distance exactly in n0111 17.

ChMn a site and integers a > m > let ft 1,1 be the event thatthere is an open path P from x to some site in S„ (:c), and let Ix ---111“ !} bethe event that there is an open path P star ting at visiting some sitey E S,,,(.e). and ending at a site c E 8,,(y) Men Ix C

m,

if P is an open path how to S„, +„(x), we may take p to be the firstsite of P in .9„,(x). As P ends at least a distance n from p, there is asite z E P. corning alter p, at distance exactly /1 from yr see Figure 1

If {:r ----1 } holds, then for some y E 5,,, (:c) there are disjoint witnessesfor the events {x y} and {y Hence,

pe(f.)?±C') 5- Pp (0 ''2") --' In {yEs„,00

PP 11)1Pe(lES,,,

E IP„(0 y)P„ (0 -Z.)y E (0)

E„(ro n S„,(0)0PI, (0 2+) S P,,(0 24-)/2,

where the last step is horn (1) It follows that P p (0 < 2-l"/"`J, so

111„ (0 -2 ) decays exponentially as nThis result was first proved by Hammersley [1957b] in the much Ettore

SO Exponential decay and el The& probabilities

11

Figure 1 An open path P from :r to Sin-Hi(r) The portion of P from :17

shores that

general setting of it 'lemon/ 9 below,incqualitN WilS protect

well before,e eauden B erg_ Kest en

4.2 Oriented site percolation

lo state the main results of this chapter we shall consider oriented sitepercolation, where the sites of au oriented graph A are taken to be openindependently %kith probability p The reason for considering orientedsite percolation is that results for this model extend immediately tooriented bond percolation : and to tuna iented site and bond percolation,by considering suitable transformations of the underlying graph Beforewe expand on this observation. let us recall some definitions

As in Chapter 4 b N a percolation measure P on a (possibly oriented)graph A \Ne mean a probability measure on the subgraphs of A Arandom element of this probability space is the open sulupaph of A: forbond percolation. the open subgtaph is formed by taking all sites of Aand only the open bonds: fin site percolation. the open subgr aph consistsof all open sites of A mid all bonds joining open sites POr a site :r of A,.

the open cluster C„. is defined from the open subgraph

2 Oriented site percolation 81.

Turning to mientecl site pet total ion. we write = P— = p forA p 4)the probability measure in which each site of out underlying oriented(multi-)graph A is open independently with probability p As in Chap--tei 1, we mite A lot the out-sabgraph of A tooted at x i.e., thesubgraph of A containing all sites and bonds that may be reached by(oriented) paths [tom x Natmally, we shall consider A as a tootedoriented graph, with toot x We write = Cri",t - for the open out-clusterof x. i e . the set of all sites y teachable from I, by open paths in ANote that a path in A is always oriented As usual in site percolation,a path is open if all of its sites ate open. We write Or (p) = 8;i:( A; p)

for the Pp-probability that Cy is infinite. and t i (y) = ti:( A p) for theexpectation of ICA

For each site x we have two critical probabilities, pin ( A: x) suptp :0,r(p) = and :37) suP{P < As Or (p) > 0 im-plies tr (p) = trivially p ( A; < psii ( A xi lu getwial, the criticalprobabilities pi' l ( A ; ,e) and Pi( A ; ,r) do depend on the site .e Howeveras noted in Chaplet 1, if A is strongly connected, i e if theia is an(oriented) path from x to y ro t am- two sites x rind y. then the criti-cal imobabilities are independent of the sites shall see later thata weaker condition is enough V\ ban the (l aical probabilities tie in--dependent of the site x. we write /4 1 ( A) and ( A ) for then commonyr/hies

In cartel to spell out in detail the connections between () limited andI mo/ iented site and bond percolation, we shall need one mote definitionLet us say that two percolation measures fps on (possibly oriented)graphs A, ate equivalent if thei e is an absolute constant A > U suchthat, whenever {i. j} = {1,2} and r is a site of A i , there is a site y ofA j and a set S of at most A sites of Aj , such that

J OC„I < PACs ] APj( U u/.4) (2)

for all n > 1 Tins definition is toughly analogous to that of equivalencefor metric spaces The reason fa consideting a set 8 of sites in thetippet hound is that, in bond percolation. we consider the open clustercontaining a given site, which is the union of the open clusters containingall bonds incident. with that site, or in the oriented case. directed frontthat site

Note that if mid Fr ate equivalent. Ind < x lot all sites 3: ofA i , then < x lot all sites y of A, (Here cotuse i\ ;„ is defined

82 Exponential decay and critical probabilities

using tire expectation corresponding to P I , and x y that corresponding to

P2 ) Roughly speaking, it follows that equivalence preserves the criticalprobabilities pH and pT

For an liner Wnted graph A. let y,(A) be the oriented graph on thesame vettex set, where each bond xy of A is replaced by two oriented

bonds, 5 and /77; Site percolation on A and oriented site percolation

on yos (A) are equivalent:- Indeed, (2) holds with A = 1: the distributions

of in the two measures PsA,p and Pt . A = ys (A), are identicalA ,p •

If A is an oriented (multi-)gr, apt', let v ) ,( A) be the oriented line-graph

of A. which Las a site for each (oriented) bond of A. and an oriented

edge from C to f whenever e and f are bonds of A with tie head of

7 equal to the tail of f The subscript b in the notation y t, reminds us

that it is or iented bond percolation on A that we shall model by oriented

site percolation on ;::, 1 ,( A )For an tutor iented graph A. let L(A) be the usual line-graph of A, with

a site for each bond e of A, and a bond joining e and ,f if c and f share

an end-vertex in A. and set v i ,(A) = y,(L(A))

It is not hard to show that, provided the degrees of the under lyinggraphs are bounded, oriented bond percolation on A is equivalent to

oriented site percolation on y;1 1,( A ), and that bond percolation' on Ais equivalent to site percolation on L(A), and hence to oriented sitepercolation on :,,,b (A) ys (L(A)) We give a formal proof of the first of

these statements

Them em 1. Let A be an oriented (inulti-)graph in which emery site

has ont-degree at least 1 and at most 0 < Pm each p E (0, 1), the

percolation meaSillt.9 PIL. and P. arc equivalent. where Cl pi)(1)A p AI p

Proof For a bond 7 E E( A ). let us write v,(7) for the corresponding

site of We couple the measures P i =and P2 = ryip by takingA p

each site of I-11 to be open if and only if the corresponding bond of A is

openFor a site 3, E A , let E(Cs)

of the bond percolation on A

c of A such that there is abond . all of whose bonds arat most Li. we have

denote the edge-set of the open cluster Cr

Inn other words, E(CA is the set of bonds

path P in A with initial site :a and finale open Since A has maximum out-degree

IC„.1— 1 IC 1E(CJI Ard

4.2 Oriented site it:notation 83

for any open cluster Cr in A

We shall show that (2) holds with A = max{A, 2,1/p}. If :e is a site

of A, let E+ (z) denote the set of bonds of A having : I , as initial siteAs an open path P in A corresponds to a path Pj of open sites inwe have

= U C..„.(7)eE (:ra

Hence,

P I (Kidd tr) P i (1E(C.J1 It — P2 U (7.:1r1EE.1(1')

As 1.6+ (x)I < A, this gives the upper bound in (2) with i = 1, j = 2.for any rt > 2 The condition for yr = 1 is trivial, as P I (ICA > = 1.

while F,OC(7) 1 > = p for any bond e of —K, and we have chosenA > 1/p.

For the lower bound, pick any 7 E (;r) Then

(1C„I ?_ 11.1 P I (1E(C,,)1 P2OC„,(7) >

It remains to prove (2) with i = 2, / 1 Let y be any site of Ai, so

y = y(ab) for some bond ab of A Then (!s, C and, whenever yis open, E(Cb ) C Cy. It follows that

pPi � II + 1) P2 (1Cy1 �. it) 5.. P I (IC„I7 n/A).

completing the proof q

The proof of the Imo/ iented analogue of Theorem 1 is similarTheorem 1 implies that 1.1'1 ( A) = whenever one of these

critical probabilities is well defined. Similarly, (X) A.(yr,( A )).Also, for an unoriented graph A. we have pl!i (A) = (yb (A)), and so

on Thus, when proving results of the type 'yr = p H ', it suffices toconsider oriented site percolation

There is a reason for the maximum degree restriction' in Theorem. 1. Ifeach bond of a complete graph K,, on It sites is open independently withprobability p, and Cr is the open cluster containing a given site x E K„,then C,.1) = e(n) as n 4x, with p fixed, while 1E(IE(C„,)1) = e(//2)•Let A be the graph formed by attaching a sequence of complete graphswith sizes nr, 1/ 2 , to an infinite path, as in Figure 2 Taking's; = 2i,

Sd Exponential decay and

tie& pi obabilities

to

Figure 2 A series complete graphs attached to an infinite path

say, we have y„,„(,A;p) = e pi21. while. A 1 (L(A); p) = e (E1pi41).

so

1/1 = (cn,(A)) A (L(A)) < dt(A)= 1/2

The construction mat he adapted to the oriented case by orienting ever y

bond 'away front :1'0 ' , 01 ienting each complete graph transitively

Returning to the general study of or Witted site percolation on a glapl/A. let us say that two sites fr and y ale out-like if the out -subgtaplis

A and A -ut rue isomorphic as tooted oriented graphs (Broadbentand ilannneisles [1957] used this tent' in fr slightly different way ) The

distribution of = Cit. depends only on A it so fin out-like sites r and

q we have

OAP) = 0a(P) a nd = \u(P)

By the out-class of a site we mean the equivalence class mulct theout-like relation that contains :if In other i\ ds, [r] is the set of sites q

such that J . and y ale out-like lAe write CT lot the out-class graph of

A. whose 1, ettices are the out-classes, in which thew is au talented edge

how [r] to fly] ii and oni if there are sites fc 1 E [x] and tj E with t'

a bond of A We allow [a] = WI, so the out-class graph may contain

loops Since the out-classes appearing among the sites in A: tr. depend

Duly on the isomorphism class of Al- and hence only on kb there is art— —edge Front Ix] to [lr] whenever xi? E E( A) lot some y' E (q]

For example, if A is E then there is a single out-class.. and the out-class graph has one vet lex with a loop If A is a rooted tree in whichsites at even levels have 2 children and sites at odd levels have 3., as inFigure 3, then there ale two out-classes, one corresponding to all sitesat even levels and one to all sites at odd levels. and in Ci -T there is an

oriented edge from each out-class to the otherMuch of the time. we shall he interested in oriented graphs with CT

finite In the unoliented case, the analogous condition is somewhat

4 2 Oriented N ile percolation $5

NAr-1\

T\

ALVJ 1;. r'' 6‘k 4 ti'AV4 VAV4

F t„ are 3 A rooted oriented I t ee in which ont-degrees2 and 3 alternate alongany directed path.

simpler: recall from Chapter I that two sites .1: and y of a graph Aare equivalent if there is an automorphism of A mapping x into y Thegraph A is of /mite type if there are finitely Malt equivalence classesof sites uncle/ this relation. In particular, vertex-transitive graphs areof finite type, with only one equivalence class. In the most interestingexamples, A will be vertex-transitive ha example, the Archimedeanlattices studied in the next chapter, shown in Figure 18 of that chapter,have lids proper H Occasionally, luwever, we shall stud y mote generalfinite-type graphs. Mr example, the duals of the Archimedean latticesother than the square, hexagonal and triangular lattices For orientedpercolation, A will almost always he such that any two sites are out-like,so ICIT I = 1 However the greater generality allowed by assuming onlythat C:v, is finite does not complicate the proofs, so we shall always workwith this assumption, rather than requiring that 1C ,T 1 =

The next lemma gives a condition on an oriented graph A that willplay the same role as the condition than an in/oriented graph A beconnected Recall that an or iented graph is .41rongly connected if. tor

any two vertices x and y, there is an or iented path from .t; to y

Lemrna Let A be an Infinite. locally finite or ienled multi-graph with

C v shrrngtroam-Ord Then there are real :umber :N (A ) and A( A )

such that

, 1( A ix) =141(7C) and itt ;

(A)

for all sites:( of A

Proof. Let and y be any two sites of A As there is an or ientedpath from [al to [y] in CH-c, there is an oriented path P in A nom xto a site it' E [y]. If P has length 1, then 0„,(p) > p 1 0 /Ay) = Oy(p),

and :y r (p) > pcx,,,(p) =p/Up). As :c and y are ar bitr y, we find thatlt,r (p) > 0 for some site x if and only if 0,,,(p) > 0 for all sites a!, andthat x x (p) < oc for some site a if and only if yx (p) oc for all sites x.Hence, the critical probabilities pli ( ; x) and 10F ( A ; r) do not dependon the site :c, as claimed q

Lemma 2 should be contrasted with the observation in Chapter 1 thatin the unoriented case the critical probability pn m fir does not dependon the initial site for any connected underlying graph A Note that thereare many interesting cases in which (.77s. is strongly connected while A

itself is not, for example, Z and the tree in Figure 3For sites x and 'y of an oriented graph A, we say that y is at distance

n from and write (kr, y) = II, if there is an oriented path from 1,

to y, and the slim test such path has length II We write S.,+, (i) fm theout-sphere of radius tl centred at i e for the set of sites y at distance

from x Note that y E St+, (x) does not imply c E S,±, (y); indeed,there may he no oriented path from y to :r at all, in which case d(y,x) is

undefined When A is obtained from an unoriented graph A by replacingeach bond with two bonds, one oriented in each direction, then d(x, y)is just the usual graph distance on A, and ,5 1,1-(x) is just S„ (A; x), theset of sites of A at graph distance n from x in A. In [act, the results andproofs in this chapter may be read for unoriented percolation simply byignoring all references to the orientation of edges

Although it so happens that such complications will not be important,let us note that the behaviour of the distance [Unction d(r, y) on orientedgr aphs tarty be somewhat counterintuitive. For example, let A be theoriented graph shown in Figure 4. Although this is perhaps not obviousfrom the figure, all sites fir A are out-like. Taking X = (0, 0), y = (0, b)

and z (2b 0, b), we have d(x, y) = b, d(y, z) = 2 b 11, but d(t,z) = b+ n

Thus, a site z may he a very long way from a site y close to x, but stillz is not far from x.

Let us write r (C„, ) for the radius of the open out-cluster C„„ i

r(C„)= sup{ : C, fl 8,1" (c) 5-4

Our main anti in this chapter is to study the tail of the distribution of(C), i.e . to study the probability of the event r(C„) > a, which is

exactly the event {:c 41 that there is an oriented path from a given site

4 .2 Oriented site percolation. 87

Figure -I The or halted graph X on 71, 1 in which two bonds leave each site(a, b) one going to (a + 1, b), and the other to (2a. b = 1) ibis example wasgiven by Paul Balistel

X to sonic site y E ,9,11Jit) Note that, as A is locally finite, (Q) is

finite if and only if C, is finite. Note also that the length of the shortestopen path from :r to 8,T, (w) may he much longer titan a

In many of the arguments, it will be convenient to wor k with a closerrelated event that does not depend on the state of X. Let i?„(,r) he theevent that there is an or iented path P front r to some site y E S,t(r),with every site of P other than :r open. This event is illustrated inFigure 5

Equivalently, R„(:r) is the event that there is an open path P' fromsome (out-)neighbom of X to a site in 8,;(3). (It makes no differencewhether or not we constrain .P' not to use the site x: if there is such apath P' visiting ,c, and y is the site after x along P' then the portionP" of P' star ting at y is a path horn an out-neighbour of 'a, to S,t(a)not visiting :r ) In the light of the latter definition, we will sometimeswrite {x + 4-} for R.„(x). We shall study the quantity

Pc ( 1, , P) = 111)s)(Bc(39)

As the event R„ (a) does not depend on the state ofand 241 holdsif and only if :r is open and R„(x) holds, we have

IF;(1(C„.) > a) = (3, 224) = piR;,(R „CO) = pp „Cr p)

88 Exponential decay and ruling, probabilities

Figure 5 An Must radon of the event U t eri Solid circles represent open sitesAs </(x. :) 4. the (unique) Open path hom the out- neighbour y of ,r to z isa witness for PIA(a)

The !casein lot considering 1?„(‘c)inter than Fa =Id. is that we shalllook tin disjoint witnesses for pmts of events such as l?„(x). fik(f),

which would he impossible lot tr 217). and Ix-LI

p < MI ( A; x), then (C5 ) < c with probabilit y 1, so Fi;,If (Cc a ) >a) — as it — oc.. Suppose flint CT is strongl y connected Then, IwJ

Lemma 2, throe is a si tgle ethical ptobability p L1'( A) independent of t he

initial site a For a fixed p < p i ( A ). we thus have p„ (:r,. p) =

lot twin site 1 7. In the i ugurnents below. w(:` shall need bounds on

( p )8111) (abli)J7E7C

obtain such a bound, we shall assume that CT. is finite.

Lemma 3, ltd A be an infinite locally finite ()nettled multi-graph with

C7v finite and strongly connected. and let p E (0 I) Then there is a

constant a > 0 sack that

/Pa( Cji, "Pit °Cul (3)

for all sites .c and p and integers ti > where IF,, denotes If

p <14 1 ( A) then

(P) — 0

US lr —

4 :2 ()hoard ate percolation 89

Proaf Since C-c is finite and sli t ()ugly connected. them is a constant tsuch that fin atIV two out-classes [r] and [y]. there is a path of lengthat most t horn [d] to [y] in CT Hence, lo t any sites .r and y. there isa path P of length t' < f from r to a site V E Let E be the eventthat all sites of P other than (pet haps) y' are open Then

1P,,(1c,„1 H) P,,(En flew] = P„Oc„ii I, F)

pl IP' pac mi I (i) = IP (I C �

and (3) follows with a =

For (-I), note first that pi l ( A ) is well defined b y Lemma 2. For p <

4 1 ( A ). we have fir, (t. It) — 0 tut each t` B u t Pik (3 . , p ) depends only onthe out: class fri of .1, Thus p„(p) is the point wise maximum of finitelymany functions each of which tends to -zero so p„(p) —

Let us remark that (3) implies 0,, > a0), and > (1\ y . with a =

the constant given by the proof of Lemma 3 It would he temptingto think that (3) holds with IC! replaced by )(Cr ) This is. in fact.true in the case of man Wilted percolation. Indeed. if (1(.r y) <

CJ

and (Cy ) > n then there is an open path P horn ir to a site z withd(y. > a. so r (C„) > > n - It follows that

p (C,,,) = ri) = p e1 P(/ (C„) (') 1 1 P(/ (Cy ) )))):, a)

In the oriented case, however, we may have (1(1 ) no t ch snuffle' thand(y. .11 - d(r. q) as in the example in Figure -I. and thole does not seemto he an ohvions reason win P(, (C,„)> n) should decay at toughly thesame rate tot different sites 3 . Alensi t ikov. Molchanoy and Sidorenho(1986] state in their Lemma 6 1 that this does hold for t he oriented case.but, their proof is for the num iented case. FOr innatel y. it nuns out that

then Lemma 6 1 is not needed in tlw subsequent arguments. since itmay be replaced by (.1) (In [1986]. Alenshikov outlined his proof for thetutor hutted case. where the corresponding problem does not arise )

Alt hough or iented percolation is sometimes hinder to troll: with thatuntalented. it so happens that the proofs in this chapter ate cqualbsimple for the oriented and tunniented models We state the resultsfin the (nitr ified case. as this implies the unoriented ease In fact. the[cadet interested onl y in the hate) case nun simphv ignore all relercucesto orientation. and so tend the proofs as if the y had been given forunor Wilted percolation

PO Exponential decay and critical probabilities

4,3 Almost exponential decay of the radius — Menshikov'sTheorem

Our main aim in this section is to present Menshikov's fundamentalresult that, under mild assumptions, below the critical probability theopen cluster containing the origin is very unlikely to be large A con-sequence of this result is that the critical probabilities pr and p H areequal Independently of Nfenshikov, Zell/M11/ and Barsky proved simi-lar results in a very different way: we shall outline their approach at theend of this section

Recall that R„(.1.) is the event that there is an open path from anout: neighbour of to some site in St(x), so { 24 holds if and onlyif is open and R„ (3! ) holds We shall star t with a study of p„(x.p) -,--1?„(R.„(x)), and how this changes with p As we shall use the Mar gulls--Russo formula the first step is to understand what it means for a site tobe pivotal for the event l?,, (x) As before, the context is site percolation

on an oriented graph A. so P p =A ,p

Suppose that the increasing event R„ (a) holds For convenience, sup-pose also that a is open, although this will riot he relevant. As R.„(x)is an increasing event, and we are assuming that R.„(x) holds, a site yis pivotal fm R„(x) if and onh if R.„(x) would not hold if the state ofy were changed from open to closed, i e., if all open paths from 37 to

Op pass through y. Let P be any open path from :r to 8",;-(r) Thenall pivots (pivotal sites) appear on P Let 1, 1 ,b,. r > 0, be thepivots in the order in which they appear along P (Our notation bt ischosen as we think of the pivots as In idges that any open path horn :rto .9;,P (a) must cross.) Note that if P' is any other open path front X to

8,i(x). then not only must P' pass through each of the sites In.

but these sites appear along P' in the same order b 1 . Ix,. Other wise,there is an s > 0 such that the order of the pivots along P' starts with

. where s 1 But then the MUGU of the initial seg-nrenb of P' up to li t and the final segment of P starting at bt containsan open path from at to S,t(x) avoiding 14 4. 1 , which is impossible; seeFigure 6

In fact, the more detailec.1 picture is as follows: the set of sites onopen paths from to 87,(v) forms a graph in which br, i br are cut-vertices; see Figure 7 Taking ho x, for 1 < i < let 7-11 denote theset of sites on open paths horn 1) 1 _ 1 to 14, excluding 14_, and b, Also,let Tr+ , denote the set of sites other than b, on open paths horn b, to.5-, f,"(x) Then, for 1 < the set T.; U contains two paths

.3 Alenshileov's Theorem

91

.5;ti (r)

Figure 6 An impossible configuration: the pivots(); appear in different ordersalong two open paths P. P' from 3. to Stec). The path P is the sit algid linesegment from to y: the section of the path P' from to 1, 1 is drawn withthick lines The path P' cannot 'jump' front I), to I), t > s+ since P UP'would then contain a path from t to p avoiding

(a)

Figure 7 The set of sites on open paths from 1 to 9;4; ( err), shown with apossible artangement of the pivots ha for the event. R„ (:r) Note that bib..LT2 63 mid br,b6 are edges of A

front 1) 1 _ 1 to h 1 that are vertex-disjoint apart front their endpoints; in

the case whole E E( A ), this holds trivially: both paths consist of

this single bond Also, T,. + , contains two paths front 1.4 to St (3,) that

are vertex-disjoint except at b r . For i there is no (oriented) bond

front a site in T1 to a site in Ti

Let ho a Whenever .11.„(x) holds, let Di , = 1,2, . denote thedistance horn hi_ i to h i , where we set a = x if there are fewer than+ pivots. Let I < k < v. Since D i is the distance from a to the firstpivot, if D I > k then either there is no pivot, i.e., there are (at least)two disjoint open paths front neighbours of x to (4 or the first pivot

h i is at distance at least k + 1 horn a In the latter case, there are two

92 Eepoio duo q and c i it Ica! probabdlLies

disjoint open paths fron t neighbouts of a' to and au open path howb i to 5 11 (r) disjoint flow T 1 and hence how these paths: see Figure S

figure S II DI > then it there are any pivots for the event 1? ” (1)• thelitst . is at disun i ty mot e than k frpm r Thus thine ate open paths P it owl

to ) and P' how S i±r) that are disjoint except at .r

l i t china case. there ate pat hs P and P' front .1 . to S,+, (r) and totespecti eI N with P 111(1 P' disjoint except at .1 and all sites of P andI' tit het titan x open illus. the event R „(x)fi { > k} is contained inI lie erect .1 „( Pk(.r) Hence. by t he ran den BEng-Rester inequality.Theorem 5 of Chaplet 2. tic lime

Pip (I?„ (3). > < Pp (R„(x))Ei,(111, Tr)),

i e .

Pp ( D I > k I Rs (l i )) 5_ 14.(1',

The wain tool in Nlenshiltov's proof is a genet alizatiou of tins observa-tion

When 1?„(r) holds and there are at least t pivots, we WI Re 1 1 lot thetilt into-1o, el u.stel meaning the set of sites z such that there is a pathP born to c with I) P and all sites of .P other than ./ open: seeFigure 9 Thus, if c is open, then I I is the set of sites teachable fron t Jrby open paths not passing through b, We use bold font fin the randomvariable I/ because it will be partic u larly impottant to distinguish Ithorn its possible values It ot her random variables, such as wedo not bother.) When lo = y and II = It , then every site in I1 U 0/1

4 :t itIen gbikov's Theorem 93

Figure 'the 2nd interior cluster Io consists of the thick hues including all oftheir endpoints except b2 Note that z 6 11 , oven though z is not on an open

path Irian i t neighbour of 3: to 51(x)

open, and all sites in d1 1 other than y ate closed. where 01, is the set of

out-neighbours of . U 1:0

With this pteparation, we me ready to present the key lemma forlenshikov's main theorem As usual. denotes the probability mea-

sure in which each site of A is open with ptobabilitv p. and the states

of the sites are independent

Lemma 4. Let x be a Ole of an unfilled !pupil A- and let 11. I nod f

 lc[ E) < supl in(y.p): C.r. A}.- (5)

where E is the event

E R„(.r) n {D i = D2 = (12. Dr-1 = -1}

and as pk(y. p)=F;,(R),(y))

Proof Throughout this proof, we work within the finite subgr aph of A

consisting of sites at distance at most n hour a We write IP„ For Fis,

We shall pat tition the event E according to the location y of the

(r— 1) 5` pivot b, and according to the ulterior cluster I= For

any possible value I of I consistent W ith E let

Ely = R.„(e) n = n =

I = then = Feu 1 < i < so the events E„ r Ionapal tit ion of E

If E„ I holds them as noted above, so does the event /7„ e that all sitesin ./u yl rue open. and all sites in RI\ {y} ale closed In tact, Ey e holds

91 ErponenHal decay and critical probabilities

it and onl y if Fr, , holds and there is an open path P horn a neighbourof y to .9,1-(x) disjoint from I U DI (Note that y E DI.) Let us writeGy.i for the event that there is such a path P

Let X = (/u0/) c . and let P IA, denote the product probability measurein which each site of X is open independently with probability p Wemay regard the event D ili as an event in this probability space. Notethat, in the measure P„, if we condition on Ft, , ,. then every site of xis open independently with probability p (The event Flo' is defined'without looking at' sites in X.) Hence,

P„(Emi F Fp .1 ) = WI;

al I ts

1Pyt.E„ 1 ) = Py (p, 1 )1? (Gm!)

Let Hy f denote the event that X contains an open path horn a nrigh-bout of y to b' (p). Suppose that E l, j holds and that D, > k ThenF;, r holds, and, as in the case; = 1 above, X U{y} contains open pathsP. P ' . disjoint except at y, with P joining y to 8,;(a.). and P' .joining

y to St (y); see Figure 10 Thus, X contains disjoint witnesses fin the

Figure 10 The shaded region represents the interior cluster I If E y Iholds, then I = I and ti is the — 1)“ pivot 14 _ 1 for the event R.,( 1! ) Iftin addition. Dr > k with d(r y)+ k < then there are disjoint paths P. P'

horn y tr,) (x) and to Si F; OIL with P. P' C = uill)'

/ illerishiltov's Theorem 95

events Cy r and H„ r. so Crl„ i q H„ r holds As Ey( n {D, > k} is a

subset of Fil l, we have shown that

P„(E„ ,1 ri{D, > Py(E, i )P;\,(C / 0 Hy 1).

Applying the van den Be t g-Kesten inequality Theorem 5 of Chapter 2,to the product probability measure P IA, WC have

F (Cy 0 H„ ,1 ) < IA, (6?„,r)Plx,(H„,r)

Combining the three relations above,

Pp (E,, ,1 n{D,. > lc}) G Pp(E„

Now for any possible ti and we have P;\ (11„ , r) = <Py (R 1,(y)) pk (y,p) Thus

1.7„ (E„ n > 0 ) 5 Pp 4E.„ 1)pk(p, p),

i e

Pp (D, > k E y 1 ) pk(q.p)

As the events E„ .r par tition E. the bound (5) follows q

Lemma -, is the key ingredient in the proof of Nlenshikov's main theo-

rem Although this lemma gives detailed information about: the distribu-

tion of the distances to successive pivots. all we shall need is the simple

consequence that, on average, there ate many pivots if 11 is large.. To

state this lemma, let N(E) denote the number of sites that are pivotal

for an event E

Lemma 5. Let .r be a site if alt leafed graph A. and let a and P bepositive integers Then

E„ (R.„ GO) 1?„(x)) HMO - sup ppOi ,p)) LikRj

Proof Let D,, D2, be as in Lemma -1 As the event

D, _ <on R.„(2)is a disjoint union of events of the form

{D, =d i , ,D,_1=61.,_,}nft.„(d),

Lemma -I implies that

Pp (D, < k . ,D, G R.„(.0) 1 - sup 14(y, p), (6)

In; Erponential decay and critical probabilities

whene\ tk < ti In pinticulin

Pp< /?„ — stIP fik (

"Faking t = 2 in (6) we obtain

iP(DI.D25 i no(.0)

r,,(D2 A . ( D I 5_ k, 17,0)) (D i 15 n„(.0)

— suPfik(il•

Continuing in this uav, we see that

li p D, < k T?„(r)) (1 —suppk(y, p))

lot ;Inv r < Link_II D, < k. then tinn y me at least t piv lid sites fin the event

!?„(x). so N(R„(0) � r Thus.

(N(R„ (tin R „CO) LitpidiPp ,D6t/tij R tt(1))

L o / k i —slIPPkOrpni-nfid

as claimed q

\Vie e BONN leafIV to puisent the tut) lundantent al tumults of 1\ letishikin[1086] (see also Nienshikm. Molchanoi and Sidotenko [1986]) The fitsishows that. undo ' n wild conditions, we have almost exponentialdeca y ()I the laditts of the open cluster below p it . As we shall see aninnuediate consequence of this result is that. lot a huge class of wind's.the t it it ical ()liabilities p i m i d pH coincide The /est/Its below tue st at edlo/ site percolation on tit iented graphs. Using the equivalences betweeniiat ions percolation models described in Sect ion 2. co/ esponding iesultslot bond petcubitkm tund for unmiented aphs follow easik

Ilene I hen. is Menshikov is main Heinen ' In this tesu. //i i ( A ) de-

notes the common value of the ethical probabilities A:1). whoseexistence is gnat ant eed by Lemma 2

Theorem 6. Let A be an locally finite °nettled multi-graphThenwith (I— finite and strongly connected and let p < p-1 1 ( A) Then thempi

Pi an o > 0 such that

EP;(., < exp(—on/(logn) 2 ) (7)

4 3 Alen sh ikon's Thew cm 97

Jot all N iles J . a ml integer s n >

The proof we present Ibllows that given be Menshikov. Molchanov andSidorenko [19861 vent/ closely The bask idea is to fix p < po < pin( A it

and to use Lemma 5 to show that at probability pa. if rt is large then theexpected number of pivots for the event 1?„(x) is large: in doing this. weuse 0(po) = 0 to show that sup p p„(y. Rn) is small lot k large Then, 6 omthe Margolis-Russo for main, it will follow that p„(x.p') = P„,(1?„(J.))

&et eases rapidly as p decreases Finall , we feed the new bounds onOre function p back into Lemma 5 Amazingly. even though we haveno a pi iot i inhumation on the rate at which sup, / pk (g, po) tends to 0as k —x appl y ing Lemma 5 again and again with carefully chosenparameters. (7) can be deduced

Proof. Since limn/ghoul ow argument 3te wort: with, site percolat ion.

we shall write for 113-7, and p it for pitAs before. for p E (Bl). let p„(p) = sup - p„(a . p) Note that bytrE

Lemma :3 lot any p < Rn ( A ) we have

p„ (h) 0

(8)

as n — xBA Lenuna 5, we have

E i (A; (R„ (a)) l?„ (J. [le / k - pk(p))1-11/bil

fm any site J. and positive integers n and k B y the Afrugulis- Russoformula (Lemma 9 of Chapter 2), for any increasing event E we Lace

T f,(E) = Ef,(Ar (E)) E (Ar (E)1 ft) = (N(E) I E)113„(6),dpso

I d

log 1111 .(L) = —P „(E) > E ,(N(E)dl (E) „(E) flp -

In par t ieulat taking E= ? „(r).

(1 log p„(x. lei I k (1 - p„(p))L”

clp

Fix p_ < p+ As ph (p) is an increasing function of p, for at 1.4]the right-hand side above is at least the value at p i.. so

Pa C r , P4-) -

Pe( 1 , P-) < xp (-(p± - p_)Lit / Li (I - p„(p+))1"Th)

og(i/pj) 5 c —J pa log(e-ilp0)

pa(7 + log( 1 /p0 ) ) 5 2p0 log(1/90 ) + pa .< (pa — p)12,

98 a:pont:Mad decay and al ilia& probabilities

As the bound on the ratio is independent of 3', it follows that

P„ ( p—)5 p„ (p+) exP — P—)[(//ki — Pb(PH-))1•"/Id) (9)

Menshikov's proof of (7) involves no further combinahnies: 'all' we needto do is apply (9) repeatedly However, it is fat horn easy to find theright way to do this..

Continuing with tire N

e proof of (7), let p < IA) be fixed horn now—on. Pick pa satisfying p < pa < pa( A). ) By (8) we have p„(pa) 0 asn Writing pa for p„„(po), it follows that if no is sufficiently large,then pa < 1/100 and pa log(l/p0 ) < (pa — p)16 Let us fix such an nohorn now On

Writing pi for p,,, ( p r), we inductively define two sequences no < n t 5

< • • • and pa > > > • by

= and p 14. 1 = Pr — pi log(l/pd,

as long as p i n > 0; in fact, as we shall see later, p i > p for all p Letus apply (9) with k = = p+ = and p_ = p1+1 Thus

1/ 1 /k) -= = pd, and pk (p+ ) = p„,(pi)== p i , so (9) gives

p ix:(m+ ) 5 P,,, / ex l) HP/ — lb+1)Ll iPd ( 1 — Pi)ri

The sequence p i is decreasing., so p i < pa < 1/100 for every i fromwhich it follows that (I — pni 1 /P , 1 > 1/3 Thus we have

p„ , (m +1 ) 5 p i exp Hp; log(1 /p i )11 /9 1 1 /3)

p i exp(— log(1/ 90/3) = p:+113

As n • • 'u, the event II,,,„(x) is a subset of H,, , (:c) for ally so

91-1-1 = (Pift) < Pa.(Pt+. 1) < Pi1/3

(

Fr om (10) we have pi < pr r so, very udely, < pal for everyAs x log(I./T) is increasing on (0,1/e),

Pa — Pi

where the second last inequality uses E i>0 )c < 1 and the final in-equality holds by out choice of no. it follows that the construction of the

sequences pi continues indefinitely, and p i > (Th -1- p)/2 ha ever y

l.3 Theorem 99

Let p' = (m p)I2 > p At this point we have constructed an increasingsequence n it and a decreasing sequence p i , such that

< Pu,(Pi)=

for i = 0,1,2, To understand the significance of this bound, weshould compare r1a and pa

Let s; = 11/Thl, so = no {L c se, and 5o = 11/pol > 100. Sinces; > so > 100, the rounding in the definition of Si makes little differenceIu particular, (10) implies that sa± i > say. Hence, for 1 < j <

we have < 4 /5) , which gives

S—,:>,(1/9)J5i--; 5 Yt

SO /11+1 = < st.inoLet n > no be 111bitutiv Then them is an i with

Iii < 11 < n i+1 = SW; < 81/10.

But

N(P) Pu t (It ) 5_ Pt:,(Pi) = < 2 / S < 2(010)-115

AS /to is fixed, it follows that there is a constant e such that

p„(p' )< ca -1 /5

(11)

for all a > 1 This weak polynomial bound is a Iar my horn (7), but intact the proof is almost: complete! All that teamin g is to use (9).. thistime in a straightforward way

Fix p" with p < p" < > 1, let k t = k i (n) = (T/5/11)Then [n/1:::] = e(n 1/6 ), and, from (11), pk ,(p1 ) 0(11 =1/6 ) It follows(1- pc,(P1))1"0''1 is bounded away front zero Hence, applying (9) with= (a), p' and p_ = ,r7

p„(p") < exp(-(p' - p")0.(nI/6)) = exp( ri I /6 )) (12)

as a oc.Fix p'" with p < p'" < p" a nd let kg = k2 (0) = ((log r1) ' (lot 2).

Front (12), we have

ph..., (lin = exp(-Q ((log n) 7 /6 )) = o(n-1),

so (1 - p1,„,(p"))1-"Thi --, 1 as a -:: z Applying (9) with k /co (n) itfollows that

p„(p '" ) = exp(-4(r1/(logn)7)).

100 Exponential decay and ci Thin! probabilitiwi

One final iterat ion now gives t he result: taking l ':n = log rt(log log O s itfollows that pfa (p'") = o(n- l ). and appl ying (9) once more we find that

p„(p) = exp(—[2(///(log n(log log n i ) s ))). (13)

and (7) follows q

The method above gives a slightly stronger bound than the inequality(7) claimed in Theorem 6 Indeed, the unappealing final estimate (13)above is stronger than (7) Iterating finthel, one can push the boundalmost to exp(--n/ log n), lint this is as far as the method seems to go.

As noted earlier, under a very mild additional assumption. Theorem 6immediately implies another theorem of Menshilrov, that > p i n, andsolr7= Pat

Theorem 7, Let A be on infinite. locally finite orientedwith C t finite and stiongly connected If there is a constant C' such that

1.5,4(01 exp(C l /r/(log 03 ) (IA)

fm fret i t i ligi then M( = Pi1(7)

Prof By Lemma 2, the critical probabilities pi i (X :3) and I/ ( A: ")are independent of the site Let p < ICH ( A ). and let .r he any site ofA Thiel'

(A : = = E 11/ n;,(// E C's) I H- ( 311firi (.1.

,r S ;':(.1 I

The final stun converges In Theorem 6, so p < pq( A ; :r) = p (A) As

P < Pi t/ "as at hitral A it 1°Ikms that (//t ( A ) Pit( A ): so 1 1/41 ( Api ( A ) q

Using the equivalence between percolation models discussed in Sec-tion 2, Theorems 6 and 7 immediately imply corresponding statementsfor bond percolation on A. and for 1 u/oriented site and bond percolation.Mann, of the most interesting percolation models satisfy the assumptionsof Theorem 7 (alter the appropriate translation to oriented site perco-lation) [ i ndeed, in many cases (Ion example site or bond percolationon Ed ), the class-graph has only a single vertex and the sizes of theneighbourhoods .51,1"(x) in A grow polk nomially iu n

It is tempting to think that for any finite-type graph, cipher the

) s Tluore tt i 101

'growth function' 2 (u) = Sup , , : ‘ 15„(.01 will be bounded by a poly-mania'. or 2 (0> exp(ur) fin sonic n > 0 Indeed. Milma 119681 con-jectured that this assertion holds in tilt' special case of Cayle y graphs offinitel y generated groups. Surprisingly. even this is not true: Gtigorchnk11983: 198-0 gave a counterexample Fut the/mo t e. solving in the nega-tive a problem of Gramm' 119811. AFilson [2004] proved that a glom ofexponential growth need not be of unifianth exponential growth (Seealso Arachnile rind Pak [2001]. Pvber 1200-11 and Eskin. Mozes and Oh[2005])

The condition that C–c be finite is essential fin Theorem 7. We illus-

trate this lie untalented bond percolation Let G i he it 2' •gt. idi c . a square subgt twit of r with 2 2' vertices. and let A be

the graph obtained b y stringing together tire gtaphs G; as in Figure II

Figure II. A series of grid-graphs strung toget he t by I heir opposite corners

lot the example, t he sizes of t he grids grow super-exponentially.

Note that A may be embedded into E 2 in a way that preserves graphdistance, so 1.5„(A; < 1.5„(72; = 4n for n > 1, and the growthcondition (IA) is satisfied If p> 1/2. then it follows easily horn the re-sults in Chapter 3 that the probability that there is an open path hornone collier of Cr to the opposite comet is bounded below by a constant(This may be shown by using exponential decay for p < 1/2 and con-sidering the dual lattice ) Also, the expected flambe! ' of sites of thatYilaV he reached by open paths horn a given cot net is 0(16l , 1) Since Pi]grows super exponentially. it follows that pi; (A) < 1/2, so. as A C V.we have I); (A) = 1/2 On the other hand. Fir (A) = 1. since. for p C 1,the probability that each of the infinitely many cut vertices is incidentwith at least two open edges is (I A slight variant of this construction

102 Exponeatial decay and critical probabilities

works for oriented site percolation: take the line graph, and then ereplaceeach bond by two oriented bonds.

In the unmiented case, we do require the underlying graph A to beof finite type. However, there is little reason to consider the graphstructure of CA, defined analogously to Cdr : the graphs A we study are

always connected, so CA is also connected. Note that if A = ips(A)is obtained front A by replacing each bond by two oriented bonds, asin Section 2, then two sites a; and y are out-like in A if and only ifthey are equivalent; in A Thus, if A is a connected finite-type graph,then C1 is finite and strongly connected For locally finite graphs,the transformation mapping an oriented or unmiented graph to its linegraph also preserves the finiteness of the class-graph Finally, thesetransformations also preserve the growth rate of the neighbourhoods, soTheorem 7 does indeed imply cm responding results for bond percolation,and for unmiented percolation

In the statement and proofs of Theorems 6 and 7, we took e ye' y site tohave the same pr obability p of being open These results extend easily toa sornew bat more general setting, in which different sites :r have differentprobabilities pd. of being open, with the states of the sites independentRecall that the definition of the class-graph C-v. or the equivalence ofsites in the man iented case, involves isomolphisms between subgraphsof A or amount ' phisms of A Naturally, we now requite any suchisomorphism to preser'e the 'weights', i e., the probabilities that thesites are open. Thus, out-like sites still behave in the same way forpercolation As we require C-Tc. to be finite, there is one probability pi

for each out-like class, so we obtain a per colation model parametrized byoa vector p = (p i , . pk ) For example, one could take A = with

alternate sites having pr obabilities p i and pi, of being open, resulting intwo oil-like classes

Theorem 6 carries over to this 'finite-type weighted context in a nat-ural way: using self-explanatory notation, the result is that if for some

. hr; We have . .pk )= 0 then, whenever pi for eachan assertion equivalent to (7) holds in the probability measure Pp,

Hew Pp , is the measure in which sites of class i are open with proba-bility flit with the states of all sites independent. There are two waysto see this One is to note that the proof given above carries over mu-tabs unitantlis In particular. the Margolis–Russo fornmla (Lemma 9 ofChapter 2) states that for tar increasing event E. the stun of the partialderivatives of IPp (E) is exactly the expected number 01(N(E)) of sites

/3 Alenskikov's Theorem 103

that are pivotal for E If we decrease each p i at the same rate, then thecalculations in the proofs above are exactly the same as for the uniformcase.

Alternatively, one can realize an arbitrarily good approximation tothe weighted model as an unweighted model, by choosing p close to 1and replacing each site open with probability pr by an oriented path oflength C chosen so that pd is within a factor p of pa,. Using this idea it iseasy to deduce the weighted version of Menshikov's Theorem from theumveighted one.

Note that we require g p i for all i: in general, it is not enoughto require p < p i for all i and pie < p i for some i. This may be seenby considering any graph in which sites of one type are ' useless ' , forexample, the graph A obtained from Z - by adding a directed cycle toeach site, where the sites of Z 2 are open with probability p h and tinenew sites with probability Po

Marry of the other results we shall present also have natural weightedversions Most of the time, we shall present only the unweighted version,and shall not discuss the simple modifications or deductions needed forthe weighted versions

Aizemnan and Barsky [1087] gave a result closely related to Theo-rem 7. Their proof uses yen v differeM methods to those of Menshikov,although it also relies on fire Vail den Berg-Kesten inequality and theMargulis-Russo for mula A key idea of the proof is the introduction ofa 2-variable generalization of the percolation probability 0(p) Consid-ering bond percolation on a graph A in which all sites are equivalent,this may be written as

moi, = 1 — E p( 7; II),rt=I

where p = 1 — 6- 0 The reason for the reparamettization is that: Aizen-nmn and Barsky consider a more general model of percolation on Ed,where 'long-range' bonds are allowed: each edge xy of the completegraph on 27,(1 is open with probability 1 — exp(-0./(x — y)), where.1 is a non-negative symmetric function on Zd . Taking It = 0, thesum above gives exactly Ilt,OC„. < = 0(p) Writing VI for

.1(X), Alain/Mill arid Barsky prove two differential inequalitiesfor AI = Al(13,11), namely

8111 < 1.111

0.111

0 8

104 Exponential dung and cIilical pi °hybrid ies

and,r

< + :11" •fli Oh Uri

The latter inegt tlitv genet alizes an eat lie/ tesult of Chaves and Chaves

1986a] that

(() < 0(i()2 0(p)i)(e)11))

lot hood pet colation on Zil (For site pet colation. the Lena 0(p)2replaced by li t 0(p) 2 ) Using these differential inequalities Aizentnan

and Barsky show that 11/(el l> ch i/2 for some c > 0. where ;IT

corresponds to p i Math the) deduce that p i / in Note that the

quantity It) has a simple combinatorial description: if we extendthe pet colation model la adding a single /KM vertex G (the -ghost'

vertex). and join each site of E d to G independentl yprobability— then 111(13.11) is the to obabilit v that thew is no open path (tour

0 to G

4.4 Exponential decay of the radius

Out aim in this section is to show t hat. undo mild conditions <I he distribution of t he 'whits of the open dust et containing a given site

has all exponential tail Floiment 6 does not quite give this. although

it conies close Flowevet exponential decav follows from Theo/tin t 7

by tesnit of Elatumetslev )19574 a special case of this result was

described at the start ()I the chaplet The statement and pool that

follow appear considerabl y tome complicated than those lin the special

case of bond petcolation on D I considered at the stint of the chapter

Then) a l e two 'fiasc 's: first. thew is a noun ' technical complication

that.vises in the case of site petcolation Second we shall separate out

the heart of the result as a lemma, and we wish to state a teasonalth

silo/1g form of this lemma ha hame 'detente If out aim we t° only to

move flanuocuslev's result. I Iwo/ em 9 below. then a weaker km to of the

lemma would suffice In tact. the much greater genetaffix consideted

here int t oduces no essential complications

Given a site t of a directed graph A. let

13'1 ( 3 ') U if ( ,I )

e 71 • (I(:r. y) :)); t

be the not-boll of r.adius r centred at Let A ,• (.r) denote the numbe t of

Exponential decay of the Indira; 105

sites in5';r(r) that ma y lie leached In an open path P iu B7( t ) starting

at all out-neighbour y of c

Lemma 8. Let A be ml oriented multi-graph r > 1 an integer. mai

< 1 a real nymber. If Elis.,(N,)+T.r)) < for every site .r of A. then

d

fat every site ;I Of A um/ CM, y 11> 1

Proof As usual. we write tr,, for IP); Let 1,n > 1 Recall that I?„, =

tr. + -14. 1 denotes the event that tin g e is an open path P From an out-neighbour of x to a site y e S,t(:r), and that a„,(r.p) = Pp(R„,(r)).By splitting the path P at the point it first reaches Sit ()), we see thatRrr.„(x) is the union of the events E„, y E .9;(,r), where Ey is the event:that there is an open path P in a,t,..„(r) horn an out-neighbour of r to87: „(:y ) that first meets .9)± (x) at q (This is not, in general. a disjointunion. since there m ay be many such paths P ) Splitting such a pathP at y. the initial segment P' Rom a neighbour of 1 to y lies in 8,5(r).and is thus a witness For the event E,C that theme is a path in 13)/(t) flowan out-neighbour of x to y; see Figure 12. The remainder, .P", of P is apath star ting at an ontmeighbout of y and ending in S iti. „(x). Thus P"must contain a site of 5,1)(q). so P'' (ot an initial segment of this path)is a witness for 17„(y) As P' and P'' are disjoint, we have

C /7„(y)

Writing p„(p) for sup y p„(y, p), as before. by the van den 3eiginecpuditv it follows that

P,±n(x, Lii)„(E„)< PpuLT;,FP,,(R„ on)

(J) itC-b;!-(x)

E IFI Y( E/y) p a Err(N;3-(I))aa(a) 5 -IM(t)

ye sr)) )

As this holds fin every we have p, ÷ „(p) < p„(p), so p„(p) < -siln/r1

lot every a As Py (x 22+) = pp„(r), the result Follows

Note that we work with paths star ting at a neighbour of a given siteto avoid -re-using' a site when we concatenate paths This complicationdoes not arise for bond percolation, where the corresponding result isthat, if the expected number of sites of 5 + (t) that may be reached

1(16 Exponential decay and critical probabilities

Figure 12 The (Wick lines show an open path P witnessing the event R.,-E.n(:c)•The sub-paths P' from it ' to y and P" front y ' to z are disjoint witnesses Forthe events E,/, attd R.,,(y)= fy + 22- 1 respectively

nom x by open Paths within B,+ (x) is at most i for every 7 then

Pi,(3, 11-) < -1 1" h -I holds for ever y kite's and integer a Mom Lemma 8,it: is very easy to deduce Hammeislev's result

Theorem 9. Let A be an oriented multi-graph with C hi finite andSilDlIgly connected. and let p < p4-( A ). Bum there is an a > 0 suchthat

11F7,(3: exp Im)

(15)

fin every s ite 3, and integer a >

Proof Let t' be a site of A By Lemma 2 we have p4 a; =

so ivj, ( p) < x Now

E - PE E -())) IIESt (x)

4.5 Exponential decay of the volume 107

where Ix + yl is the event that them is an open path flour an out-neiglthour of x to y (Thus, I?, (x)al Also, as

before, if {x + y} holds then there is an open path fiom (x) to y

not visiting: , so the events is open' and {x± --r are independent; )

= Uaes;t-cofx-i-

Let

"o(c)= E r+Pacis;•(3,)

Then E r 7, (x) is convergent for each x, so 7,.(r) a As 7, (a:) dependsonly on the out-class [x] of x, and there are only finitely many out-

classes, there is an r such that 7, (x) < 1/2 for every site x Since

4,(N,+ (x)) < (x), the result then follows how Leman 8. q

Let us remark that, under the assumptions of Theorem 9, if the ex-pected untidier of open paths star ting at each site x is finite, then (l5)

can be deduced without using the van den Bet g-Kesten inequality. Flow-ever, it is pet reedy possible for this expectation to be infinite even whenx =i,(p) is finite, as shown by the graph in Figure 13 Indeed, for this

7N 7N 7N

N7 N/Figure 1$ A graph With = I in V.111(711 there are exponentially many pathsof length C from a given site.

graph Pit = = 1, but taking each site to he open independently withprobability p, the expected number of open paths of length C starting

at xo is ,ri which tends to infinity as C. for any p > lb,F5

Performing the saute construction star ting with a binary tree rather thana path gives a similiu example with pit = pis! < 1

4,5 Exponential decay of the volume - theAizenman-Newman Theorem

Om aim in this section is to strengthen Theorem 9: under suitable con-ditions, not only does the radius of the open cluster . C„, containing a

given site a: decay exponentially, but so does its ' volume ' , IC1I Aizen-man and Newman (19811 gave a very general result of this form, winchwe shall come to later However, a special case of their theorem may

108 Exponential decay and critical probabilitic5

he proved in a ver dilleren t way, using 2-independence: we present thisHest

shall for initiate the next result for until iented site percolation ongraph A Recall that, in this context, two sites are equivalent if therean autorumphisur of A mapping one into the other (This corresponds tobeing out-like if we replace each edge by two oriented edges ) A graph isof finite type if there are finitely many equivalence classes of sites underthis relation As usual, we write S, (c) For the set of sites at graphdistance a from a \VC write B„(x) for U7--u Si(/1 ) , e for the hall oradius a centred at a'

The conditions in the result below are generous enough to ensurethat it applies to the 1/10tit hale'fir g tiltieS, including site and bondpercolation on D I and on Atchimedean lattices The proof is basedon Theorem 6 or Theorem 9, and the concept: of 2-independent sitepercolation Although it is ratlwr easy, and strongly ierniniscent of thepool . of Theorem 12 of Chapter '3, hem we have to soil( hauler to geta covering corresponding to the cover of 2 2 with large squares

Theorem 10. Let A la a connected, infinite locolly-fruityfinite-type

wan tented graph S l ippage that

sup 12,011 < t (hig t Vinon

(16)

fa all soffit amity tangy t net fat meet// p <

o = u(11 p) > 0 such that

;MCH > a) < exp(—on)

Lot all sites . t and all > 0

/)11(-‘) ill( I

Proof The (nem!l plan of the proof is as follows: NW shall cover 11(A)In a set of balls Bo, OIL u E II' that are reasonabl y \\ ell spread out':binning a graph (F) on II" by joining a'. a' ' if they ale at distance atmost tit . say we shall show that the maximum degree of D(11) is nottoo large We then construct a 2-independent site percolation measureon D(II ) as follows: a site la E II will be active if some E gi,(w) isjoined In an open path to a : distant' site i e a site at distance at leastr limn .r: these events are independent if d(W. a') > 6/ Also. an y sitein a large enough open utast:el is , joined to a distant site. so a large opencluster in A implies a l a rge active cluster in D(IJ ) From Theorem 6(\lenshiky . 's rliemein) and inequalit y (16). each a' is yen in /likel y to

4 5 Exponential decay qf the volume 109

)e active if t is chosen huge enough: it will then Follow lion Lemma I

of Chapter 3 that open clusters in this 2-independent measure are small,giving the resultc, -

To carry out this plan, we shall first show that balls of /minis Tr inA ale not too much huger titan those of radius r Let A < c be themaximum degree of A As A is connected.. (di each pair ([4. [y]} ofequivalence classes of sites thew is a path P hour some E Ix] to some

E [y]. Thus there is an integer L such that for am .r and (1 , there are

E and ij E (y] with d(3: ( (/) < L. It follows that. crudely.

i prODi = i 13,Cri )i i r3 ,+1.0a = (I +A+ ±AL)18,,(y)l

for every t > I Let left' = ntaxilB, :.r E A} and bi = :A}. Then we have I < < C' for every where C' depends

only on A.\\'e claim that

< (18)

holds Mt infinitel y many t Otherwise. there is an rn such that 1.11> t 0 we hate

let 11/:" ) > \/7: IC'

Tin t s. lin r large enough, adding log? to = log y increases log/; byat least (1/3)logt = (/3, which implies that log ft- > 1 2 /100 lot huget . contradicting (lay and proving the claim F l om now on we consideronl y r for which (IS) holds

Let C 1'(A) he a maximal (infinite) set (}1 sites subject to the halls(w) : w E 111 being disjoint (Such a set 11' exists in am graph

Lv Zonfs Lemma In fact. since A is connected and locally finite, Ais countable. so IV may be constructed step by step ) Note that theballs {Tin, (w) : w E It } covet V(A): if sonic site y (lid riot belongto Wc it -8.), ( 11 ') . then q could have been added to IF. We define anauxiliar y graph D(11') with vertex set 11 - as follows: two sites W. a E H -ate adjacent in D(11 7 ) if d(w. n) < . where t1(.t.. te) denotes the graphdistance in A If r„ denotes the set of neighbours of 0 , in D(11"), thenthe balls (13,06 : u ate disjoint subsets of 137 ,(w): see FigureHence

lb,- 5_ Li T3, (u) < [37, (o . )1 <Er„

Therefore. using (18). we have bt, < vT. the maximumdegree A i of D(11") is at most \AT

Figure H. Balls of radius i centred at a' and

variot ites a c Hiehalls are disjoint as (w} U F t, C

Let us say that a site w E IF is active if there is at least one sitey e B0 1 ( IV) for which {y L} holds. Otherwise, w is passive, Note thatthe disposition (active or passive) of w E IF depends only on the states(open (A closed) of ri t e sites of A within distance fir of w Let U and U'be subsets of IF such that no edge of D(1I) joins U to U'. Then no siteof A is Ay ithin distance 4, of both U and U', so the dispositions of thesites in U ale independent of the dispositions of the sites in U' Let Pbe the site percolation measure on D(OT) defined by taking the activesites to he open What we have just shown is that P is a 2-independentsite per colation measure..

The probability that a given site in IF is active is at most it,sup s 1F'7) (y 1-' ) if r is huge enough then, from (16) and 'Theorem 6,

we have

pt (21 )log(2, yam exp(—r Nog/ )2) exp(—r2/2)

To apply Lemma 11 of Chapter 3, let us define

[(I) = imAt-to

Since = A(D(11 1 )) < whenever (18) holds, we have

f (r) < exp(-1) /64- 0 ( )

4. 5 Exponential decay of the volume 111

In particular, j(r) < 1 if t is large enough, and, as (1$) holds for in-finitely many I , there is some r for which (18) holds and f ) < 1 Fromnow on, we fix ;Men an .

Applying Lemma 11 of Chapter 3 to the 2-independent site percolationmeasure IP on the graph D(W), we see that there is a constant c > 0such that, for any z E IV and any nr > 0,

i13 0C.(D(11 7 ))] in) 5 exp(—cm),

where Cz (D(IF)) is the open (active) cluster of DO V) containing zTo complete the proof, note that if x is a site of A and I CA >

then for every site y E C„, the event {y =} holds. (There is no roomfor all of Cr inside B, _ i (y)) Let IV' = fw E FP : (w) n 01Then every site w E IV' is active Also, as the balls Bo, (w), w E TV,cover 1 7 (11), the balls Bo, (w), E IV', cover Ci„

We claim that 11 7 ' induces a connected subgraph of D(11) Indeed,if w, E then there are sites y, y' E with y E .80 1 (w) and

yj E (al) As C'„, is connected, there is a path y = !pry° • • yt =with every y i in C . We nilw choose Iv( E IV' with Eh E Bo, (wi),and to, = w, air = iv'. For 1 < i < t — 1 we have death<

= 4r ± 1 < 6r, so either mi = witvi+, is anedge of D(IV) He/we, w and iv' are joined by a path in the subgraphof D(11") induced by IV'

Fixing E A, let E II be any site such that :r E B2r (2) We haveshown that II' is a set of active sites that is connected in D(II), so TV'is a subset of Cz (D(W)).. Also. as C . is covered by balls of radius2t, we have > Hence, for n >

> n) < HA )

Ill'OC',;(.0(1V))1 � n/q) exp(—cnnib )

As r is constant, the proof is complete q

In the p oof of Theorem 10, we used Menshikov's Theorem. Theo-re 6, to obtain a good bound on ner 4rn .) for p < Mi . Instead, wecould have used Hanunersley's much simpler result, Theorem 9. Mod-ifying the proof in this way would give the satire conclusion, (17), butonly for p < . (Of course, under the assumptions of Theorem 10,

= p by Menshilcov's 'Theorem )

Next we shall present the general result of Aizernnau and Newman[1984] mentioned at the beginning of the section.

El

1r

I 12 Exponential decay and critical probabilities

Theorem 11. Let A he infinite lacally-finilc oriented multi- graph with

r71111( and strongly connected. and le! p < pq ( A ) Then th in is art

o > 0 such that

El ;(1Cli I ir) exp(—on) (19)

fat all sites re and integ ers n > 1

Pox"' As p < pl.(71.) is fixed vve suppress the dependence On pin ofnotation

Suppose first that Csc has only a single vertex, i e that all sites arcout-like this is the wain par t of the proof: the extension to [-hadmany out-classes is a minor variation

As berme let fr y+ plr denote the event that there is an open pathfrom an out-neighbour of x to y interpret {.r+ — .r} as the eventwhich always holds. Thus — It holds if amid only if .1' ± — II and :1' isopen. Note that

PLr li E P(.1. !t)/P= yr(r)/P

pc., ye:\

\\ le shall estimate the moments of ICH in terms of using the van den

Berg rkesten inequalit \ start with the second momentSuppose that .yr. E We mk, not assume that z. fir and yo ate

distinct. although the Heinle is clearer if we do. As um E thereis an open (oriented, as always) path Pr horn x to yr Let Pr, he mshortest open path how a site a on Pi to /12 : such a path exists as

/12 Then Pr and Pr are vertex disjoint except at a (otherwise. Prcould be shortened) Let us split PI into two paths. a path P; horn a:

to and a path Pi/ horn a to yr: see Figure 15 ()witting the initialvertices horn the paths P(. P(' and P, gives disjoint witnesses for theevents IT + --, — I/1 { -- !P I }* respectively.Tills ' if1./2 E Cr. Olen the event

- 0 {a + IL}

holds Ica some tt E A (Recall that 0 is an associative operation) Usingthe can den Bet g- IKest en inequality, it follows that

FOP fig ) < E - it)P(u + — 111)P(a + r12)

4 5 Exponential decay al 11u; voleuntc

113

Tigre 15. Open paths Pr = Pi U and P2 showing that In t E Alterfelering the initial sites fl ow 12;, Pi" and P2. these paths are disjoint

Stunninig wail, E A, it follows that

Eacid 2) = Ey; PO/I, /12 E

gt 112

E IPI (r: 4) — f () POI'" — :01 )Pfli+

(/1

where, hour now on. all summation variables rnu (wet all sites of A

As all vertices are out-like, we have E(L+ a7) = foi any

• E A Evaluating the triple stun above In first summing over yo, Own

over yr. and theft ()VC/ it, it follows that this sum is Eractig (‘T. so

E.(1(7,1 2 ) (‘')"

For higher moments the argument is ver ybut slightIN battler

to write down: the only additional complication is that if in.. E Cif

and Pt , Pi and v t = rt are defined as above. then if ya is also in Cad the

shortest path P3 from a site E Pr U A to ya may start limn a site it7,

on P(, on Piu , or on 12, It will be convenient to write tin fot and todenote the 'branch vertices' by

We shall say that an oriented tree TE on the set

+ 1 {0, 1, 2. . k, —1, —2. .—(k- 1) }

is a k4ettiplate if it satisfies the following recursive definition: for k = 1,the only 1-template is the tree consisting of the oriented edge 01. For

> 2, the tree T is a k-template if it way be obtained flout some (k ' - 1)-

template r by first inserting the vertex —(k — 1) to subdivide an edge of

T. and then adding an oriented edge nom —(k — 1) to k Note that the

number of k templates is exactly A'-t; = 1x3x5x x (2k-3) = (2k —3)!!,

as there are c(T') = 2k — 3 choices for the edge of to subdivide..

111 Erponential decay and critical pinbabilities

Fixing the site = I/o throughout, a realization 1? of a template T

is a sequence ,:yk, . y_ k + i of (not necessarily distinct) sites

of A such that there are disjoint open paths P E E(T), withu

a (minimal) witness for: yt yj; see Figure 16 We call yr,. line the

SI

Fignie .16 A realization of a Ttemplate The directed open path from

2 to MI is a witness for Olt.,

leaves of the realization Note that we Mal' have y; = yj ford 0 j, in

which case Pr, is the 'empty path' with no sites or bonds

Recalling that yo = we claim that, whenever yr, sic E Clz , there

is a realization 1? of some template T such that 1? has leaves yr,The proof is by induction on k. For k = 1 the claim is immediate: as

Y./ E Ca, and x yo, them is a witness for {fil" yr}, which is all that

is required For the induction step, given a realization 1?! with leaves

, yk _r, let P be a shortest open path from a site y_ k+ 1 appear ing

in RI (i.e from a: OF from a site in some P-r, in /V) to yk Splitting

the witness (path) on which y_ k+i appears as in the case k = 2 above,

and taking P (without its initial vertex) as the witness P for(--k+l)k

fy k+1 yk l, we find a realization I? of some template T

By the van den Berg-Kesten inequality, the probability that T has a

4.5 .E.eponential decoy of the volume 115

realization 17. corresponding to a particu lar sequence is

P {Y; 11)}) 9.0E

Hence, the probability that yk E Cr is at most

EE77E1

Pellif. 1/k ),

where denotes summation over k-templates, and tnunation

over sites y_r, , E A Thus, summing over yr, yk.

E(Crik)

(T), (20)T

Wile/ e

;r(T) =

II — Yr).75E/

with the sum r.r- running over sites ,;(p„ ,y_kr1.1AT

The definition of 7r(T) may be extended to define 7r(U) tot any or ientedl abelled tree U with 0 as a vertex: we sum over a variable rki for: eachvertex i r 0 of U. It is easy to see that if every edge is oriented awayfrom 0, as is the case it U is a template, then

.R(u)=(x/)`gn

(21)

irked there is a leaf j 0 of U. so ij E E(U) for some i.

P(tI; - )

(22)rer

for any we have 7(U) =x1 )-r(U 1 ) where U' =U-ij. and (21) followsby induction

Rom (20) and (21), we have

EaCzik) (.0r2(7) = Nkm2k = (2k - 3) ( ))2k-1

for every k Hence, for any t > 0 we have

(2k - 3)H , k

( i (A )-) ALT (exp(tIC, I)) = E (t. k I Pr:!)

116 Erponentiol decay and el Then/ pinbalnlihes

As (2k -a) /1,7! < 2 k . this expectation is finite lot 0 < t < (‘')- 2 /2 As

E(exp(ird)) - exp(tn)PaCr i = n),

it follows that 1.11(rid = 11) < exp(-/o) fin all sufficiently huge 11, andthe result follows,.

SO fat. we assumed that all vertices ate out-like As plot/Used, ex-tending the result from this case to the general case is sttaightfonvatdIndeed, the pool is the same, except that we 'enlace by

\" = SUP

P UP . — In= SUP \if (v)I p

In place of 22), we then have

11) (lif (b) <! j

so rr(U) < (\i ")'“)) . As shown in Lemma 2.. when Eli is strongly con-nected, then \ „,(p) is finite Mt one site w if and only if it is finite finall sites. i e if p < Emilie/ mote, as ‘„,(p) depends only on theout-class of W and (7- .c is finite, the supremum above /WO' he taken mera finite set. so \i " is finite The test of the proof is unchanged. q

Let us recap very hiiefly the main themenis of this section; all theseexults matt n finite-t ' pe graphs with st rough- connected class-g t aphs.

We have moved results of Hammersley and of Aizenman and Newmanthat.. nuclei mild conditions, for p < p T < p it , both the radius and theVOIIIMP of the open cluster containing a given site decay exponentially.\\M have also moved Menshikov's Theounn that, under a very weakcondition on the growth of the neighbourhoods, the critical probabilitiespi and pu t coincide. Thus. : fin a very wide class of pound graphs,

p u t pun. and if p is less than this common value p, then the size of theopen clustet of the origin decays exponentially.

A very interesting problem that we have not touched upon is the speedof this decay. Mo t e pecisely, what me the best constants ca in (1.5) andin (10), and how do these constants depend on pas it approachesflow below. The arguments above give some bounds on the constantsin terms of other quantities, butt then behayiuur as p pr is a \tidifficult question. about which rather little is known We shall retain tothis Welly in Chapter

5

Uniqueness of the infinite open cluster andcritical probabilities

Ou t hist aim ill this chaplet is to plesent a result of Aizemnan, Kristenand Newman [19871 that, under mild conditions, above the critical ptob-ability is a unique infinite open cluster; Button and Keane 119891have given a very simple and elegant proof of this result Together withMenshikov's Theorem, this uniqueness result gives an alternative poolof the Harr is-Kesten Theorem; this proof is easily adapted to deter minethe critical probabilities of certain other lattices. 'The key consequence]of uniqueness is that, under a symmetry assumption, the criticalabilities for bond percolation on a planar lattice and on its dual muststun to I \\ re shall prove this, along with a corresponding result forsite percolation, assuming only ordei two symmetin y Finally, we discussthe star-delta transformation, which HMV be used to find the criticalprobabilities for certain lattices that ale not self-dual

5.1 Uniqueness of the infinite open cluster – theAizenman–Kesten–Newman Theorem

'Throughout this chapter the uncle t lying graph A will be Imo/ iented„ andof finite-type In CAIRN' words, there ate finitely many equivalence classesof sites undo the relation in which two sites :r and y are equivalent ifthew is an auton torphism of A mapping y, in which case we wine3' 1') j As usual, A will be infinite, locally finite, and connected

\Ye stint this chapter with the result of Aizemm t , Kesten and NOW-

WWI 119871 that, in this setting, with probability 1, there is at mostone infinite open cluster For the special case of bond percolation on..,•,,. was proved by Harris [1960]; Fisher 11961] noted that Harris'sargument carries CVO/ to site percolation' on Z 2 A little later, Candolfi,(trimmed and Russo 119881 simplified the original proof of Ali/TIM/MI.

118 Uniqueness of the infinite open du tiful and critical probabilities

Eesten and Newman. A very different and extremely simple proof wasthen given by Bur ton and Keane [1989]; we shall present their elegantargument below Related results for dependent measures were provedby Gandolfi. Keane and Russo [1988] and Gandolfi [1989].

This section is the only place where we shall use proper ties of infiniteproduct spaces in a non-trivial way (although, of course, we could re-write even these arguments in terms of limits of probabilities in finitespaces if we wanted to)

Recall that the probability measure 111/, P;\ 4, is a product measureon the infinite product space Q = {0, 1} F(A) .. Thus, the a-field E ofmeasurable subsets of Q is generated by the cylindrical sets.

C(17,0)= ful E Q : rdf o- f for f E F1,

where F is a finite subset of i t (A) and a E {0, 1}FAs usual, an event is just a measurable subset of Q Thus, c.ny property

of the open subgraph C) that depends on the states of only finitely manysites is an event, and countable unions and intersections of events areevents In fact, any property of 0 that one would ever want to considerin percolation is an event. For example, recall that {x 1..} denotes I heproperty that there is an open path flour the site 3, to a site at graphdistance I/ from 3: For a given site x and integer ft, tins property dependson the states of finitely many sites, and so is measurable It follows that{:e x} is an event Similarly, 'there is an infinite open cluster' and

her e are exactly two infinite open clusters' rue eventsAn event E is autamarphism invariant if, for every antomorphism y

of the underlying graph A. the induced autornorphism y'' : 12 Q mapsE into itself. hr particular.

= there are exactly k infinite open clusters 1

is an automorphism-invariant event for k = 0,1, oc.,. The only prop-erty of automorphism-invariant events we shall use is that any antomm-phism-invar hint event has probability 0 or 1 We state this as a lemmafor site percolation The corresponding result for bond percolation fol-lows by consider ing the line graph, noting that if A is of finite type, thenso is L(A)

Lemma 1. Let A he a locally finite, finite-type infinite graph, and

let E C Q = 01'0) he an antonunphism-invariant event Then

In p (E) E }0,

5 1 The Alzenman-Kesten-Nenynan Theorem 119

Proof We shall write F for FA 1, Let xo be a site of A Note that, asA is infinite, locally finite, and of finite type, there are infinitely manysites x equivalent to to

Let > 0 be given Since E is measurable, there is a finite set F ofsites of A and a cylindrical event Er depending only on the states of thesites in F. such that

P(ELEF) < E (1)

Let AI = max{d(xo, q) : y E F} Since A is locally finite, the ballBom(xo) = {z : d(x0 ,z) < 2A1} is finite Thus there is a site x with

xo and d(x,,r0 ) > 2.111 Let (p be an autonnuphism of A mappingxo to a. For y E F we have

d(te,y)(0) d(au, Y0'0)- d(V( to), (PM= der° , 37) - (1.070 ,:y) > 251 - AI =

so cp(y) F. Thus the sets of sites F and y(F) are disjoint It followsthat the event y(EF ), defined in the natural way, is independent of ErSO

NEL n v(EI )) = F(Er)F (y (Er)) =INET )2

Thus,

1P(E) F(Er)2I = 1.P(E n E) -P(Er n ))I< P((E n E)A(Ef, n y(EF )))

For any sets A, B, C. D we have (Anr)A(c C (Aa,C)U(BLD).Thus, as E is automm phisrn-inwniant,

I P ( E) - P(Er )2 1 < F(EL1EF)+NEOV(Ep))

P(ELEF)+PP(E)6,y(EF))

= F(EL& ) P(EAEF)

= 2P(ELEF)<

where the final inequality is from (1)

Since 1P(Er) - F(E)1 < IP(ELEF)1 a we have

I F ( E ) - P( E ) 2 1

< 1P(E) - P(EF . ) 2 1 + ir(EF)2— P(E)21

< 2 ± 9E 45.

Since s > 0 is arbitr � it follows that P(E) = P(E) 2 , so F(E) is 0 or 10

iitji

120 Uniqueness of the infinite open cluster and critical pivhabilities

In the proof above, we did not need E to be invariant uncle/ all an-tomot phisms of A. just under a set of autornmphisms large enough thatany finite set can be separated horn itself by such an autornmphisinIn the context of lattices in R d . for example, invariance under a singleautomm phism y of A conesponding to a translation of IF'' thiough anyvector (a l , ati) (0,0, .0) is enough. In particular. Lemma 1is often stated for translation- invariant events In the terminology of er-godic theory, we have shown that the induced automorphism : R Qis et godic.

NAre now turn to the particular event lk that there are exactly k openclusters, where (1 < k < x We star t with a lemma of Newman andSchulman 119811, showing that, with probability 1, there are 0, 1 orinfinitely many open clusters

Lemma 2, Let. A be run infinite locally finite. finite-type graph. and le

E (0.0 Then

PA , = ft\:<k<x

Ilene( Why IF p (10 )= 1 Ps.% ) = 1. tit ) = 1

P100.1 By LC/111/18, 1. it suffices to prove the first statement: As before,we write for

Suppose for a contradiction that P(/(,.) > 0 fon some 2 < k < c LetTo be any fixed site of A, and let be the event that lk holds, andeach infinite cluster contains a site in B„(x0 ) As the balls B„(ry) coverA. we have Lk = u n T, k, so P(Ty ,/ P(11,), and Oleic is air 11 suchthat P(T„ , r) > 0

Changing the state of every site in B„(,ry) to open, we see that P(./ 1 ) >0 Indeed, spelling everything out in great detail, the event T„ A is thedisjoint union of the events

=T„ .:n{5' = s}.

where s (s„);,,Bu(,,„,) E k (13" 1 " ;) , and 5' = (5,.):,..EBu(,,„) is thevector giving the states of all sites in

(to) Thus there is an s for which

P (-rs k s) > 0. Now if w E T„ k s and w' is the configuration obtainedhorn w by changing the state of each of the closed sites in f3„(ro) h omclosed to open, then w' E In Thus

P(1 1 ) La ( {if : E T„ = (p1(1. p))1(TT„ s ) > O.

5 I The Airiemnan-Aesten-Neuunav Theorem 121

where c is the number of sites in .13„(:ro) that are closed when S =s

We have shown that if P(//,,) > 0 for some 2 < k < cc. , then P(I,) > 0

But then P(4) = P(1 1 ) = 1 by Lentela 1 As 1k fl i t = 0, this isimpossible q

To prove the main result of this section.we shall need a simple deter-inistic lemma concerning finite graphs

Lemma 3, Let C; he a finite graph with k components and let L and

C = , es } he disjoint sets of ITT bees of C. with at least one et

in each coinponent of Let . be integers each at least 3Suppose that fin each i deleting the reflex m disconnects the component

containinf«. 1 info mullet components. in; of which contain mudd ies of

L Then

ILI > 2k 4- E(In i — 2)

Proof By considering each component: of G separately, we 11/111/ assumethat k = 1, i e that C is connected Flemming an edge from Ce canonly increase tire number of components of — e i containing elementsof L.. so we may assume that C is minimal subject to CI being connectedand containing C'U L Thus C is a tree all of whose leaxes are in CUL

As > 3, no vertex E C' can be a leaf of C. so all leases are in L

(there ma n also he internal vertices in 14

Since C — has in; components, the vertex has degree in; Butfor any tree with at least one edge, the number of leaves is exactly2} 5 (d(e)— 2), where the sum HMS over internal vertices. As d(v)> 2for each internal vet tex, the sum is at least. 5-7= ,(nc —2), and the resultfollows. q

Let us say that an infinite graph A is untenable if 1.5',,(r)iABflOpi —'as n for each site 3, i e . if large balls contain man y more sitesthan their boundary spheres. There me several notions of amenabilityfor graphs; in the present context this \In hint seems to be the mostuseful. In fact, the concept of amenability originated in group theory,where it is defined somewhat differently. Note that. if A is amenableam( of finite type, then the limit above is automatically uniform in r.

The following result is dire to Aizemnan, Nester' and Newman [1984the proof we shall present is that of Bruton and Keane 119891

Theorem 4. Let A he a connoted. locally finite. finite type. amenable

122 Uniqueness of the infinite open cluster and critical probabilities

infinite graph, and let p E (0,1) Then either 1FX p (I0 ) = 1 or rA p(B)

1, where Ik i s the event that there are exactly k infinite open clusters in

the site percolation WI A

Proof As usual, let us write P for P;\ p In the light of Lemma 2, thisresult is equivalent to the assertion that P(/,) = 0. In fact, we shallshow that the probability that there are at least three (possibly infinitelymany) infinite open clusters is zero Suppose for a contradiction thatthis is not the case.. Let to be any site of A, and let X0 be the set ofsites equivalent to zo As the balls Br (ra), r > 1, cover A, there is an psuch that, with positive probability, Br (x0 ) contains sites From (at least)three infinite open clusters. For the rest of the argument, we fix suchan I

Let Tr (_y) be the event that every site in B, (x) is open, and there isau infinite open cluster C) such that when the states of all the sites inB, (a) arc changed horn open to closed, 0 is disconnected into at leastthree infinite open clusters; see Figure 1 If w is a configuration in which

Figure I A site a for which Tr (c) holds, Ever y site in B, (r) is open; the restof the open subgraph is shown by solid lines If the sites in Br (x) ale deleted,or their states changed to closed, then the infinite open cluster C) falls intofour pieces, three of which are infinite

B, (to) meets at least three infinite open clusters, and w' is obtainedfrom w by changing the states of all the sites in B, (to) to open, then

E Ti (a0 ). Hence, IP(Tr (to)) > Thus, for all sites x E X0 we have

P(T, (a)) = a. (2)

for some constant a > 0Our next aim is to sinus that, if /I is much larger than 1, then we can

5 I The kicennion-ICesten-Arcoonon Theorem 123

find marry disjoint balls B, (:r), :r 6 X0 , inside the ball B„(:to) In fact,to make the picture clearer, we shall find balls B, (.r) that are far fromeach other. To do this, let; If c X0 n B„_, (x0 ) be maximal subjectto the ba lls (c), w E W. being disjoint If lo t E Xo n B„_, (x0),then (1.04/, w) < 4t for some w E otherwise. w' could have beenadded to If . As A is connected and of finite type, there is a constantC such that ever y site is within distance C of a site in X 0 Thus, everyy e B„_,—a:ro) is within distance t of some w' E X0 n B„_,.(3,0), andhence within distance 4t+t of some w E W In other words, for a > +11,the balls BID-4w), w E H. cover B„_, _ ( Gro) Thus,

IH1 1-812-1-(edi/P-1,-H:(34

ver y crudely.

1B„,n(x0 )1 ,5_ IB„_,_ 1 0. 0 )10 + A + A 2 + A'±t+/),

where A < x is t he maximum degree of A Thus. since I is fixed, thereis a c > 0 such that

(:co)i

for all n > t C. As A is amenable, it follows that

111 7 1 > a -1 1,9„ + ("di

if n is large enough, where a is the constant in (2) Let us fix such an aLet us call a ball B, (w) a cut-boll if w E 11 7 C B,,-, (::0 ) and T, (w)

holds. Note that if Br (a.,) is a cut-ball. then B, (iv) C B„ (co), and everysite in B,(w) is open Since w ti ,r0 for every w E by linearity ofexpectation ' the expected number of cut-balls is

E P(T,(w)) = 011171 IS„.H(370)1„'Em

Hence, as P(Z > IE(Z)) > 0 for any random variable Z, there is aconfiguration w such that, in this configuration, we have

s 18n-m(tie)1, (3)

where s is the number of cut-balls As we shall soon see, iris contradictsLemma 3 For the rest of the argument, we consider one particularconfiguration w for which (3) holds: in the rest of the argument Uncle isno randomness.

Let denote the union of all infinite open clusters of the configurationEL) meeting B„ (4), considered as a subgraph of A. In the configuration

124 Uniqueness of the infinite open cluster rind ci Weal probabilities

w' obtained from w b y changing the states of all sites in cut-balls toclosed. the (perhaps already discotmected) clustet 0 is disconnected intoseveral open clustels, some infinite and some finite Let the infinite onesbe L 1 , L 2 . ,L t , and the finite ones Fu Each L i contains 1site in 5'„+1(:ro)„ so

TIM/] (-I)

Let the cut-balls be CIL, C. We define a graph H from 0 by con-trading each cut-ball Ci to a single vertex each F4 to a single vet tex

and each L i to a single vet tex en see Figure 2 In the graph H. thereis an edge how Ci to ci , for example, if and only if some site of fq isadjacent to sonic site of CI;

Infinite components of (9 cot espond to components of H containingat least one vertex in L = Tints, the condition that Ci is acut-ball says exactly that deleting c i Flom H disconnects a componentinto at least (Mee components containing vet tices of L. Thus we /nayapply Lemma 3 with in; > 3, i = 1.2, „ s, to conclude that

t= 2 ± E ( 3 – 2) = s + 2.

This cont /adios (3) and (- ). completing the wool q

Simpl y put, Theorem 4 tells us that, above the critical probability/4 1 . almost smelt- theme is a unique infinite open clustet TO concludethis section we remark that, by a simple argument of van den Beta andKeane (198 .1], Theorem 4 implies that 0(1 .,p) is a continuous function ofp, except possibly at p =14[(A)

5.2 The Harris–Kesten Theorem revisited

Combined with Menshilrov's Theotem, Theorem -t leads to vet anotherproof of the thuris-Kesten result that pl'i (Z2 ) = (22 ) = 1/2 Thismoot will adapt easily to give the exact values of the critical ptobabil-ities tot certain tithe/ planar lattices We start by improving the 'ease'inequality that p li),(Z2 ) > 1/2 Mote precisely, we shall deduce Mattis's'McGloin, restated below, nom Theorem -1 The mg:tin/eat: we give isdue to Zhang; see Ctimmett, (1999, p 2891

Theorem 5. Fm bone! percolation cm E 2li am 0(1/2) = 0

5 :2 The Harris taster Theorem revisited

125

Figure 2 The upper figure shows the union 0 of all infinite open clustersmeeting B„dro) The shaded halls, in which all sites are Open, are the cut-balls. The lower figure shows the corresponding graph H The filled circles ar ethe vertices e i corresponding to the cut: balls, the hollow dines the verticest i corresponding to the infinite clusters L t , and the crosses the vertices dicm esponding to t he finite clusters

Hive( Suppose not. Them applying Theorem -1 to site percolation on

the line graph of Z2 we see that iP itql ) = 1 . where I I is the event that

126 Uniqueness of the infinite open eluslel and critical pmbalatities

there is exactly one infinite open cluster, mid It followsthat there is an no such that, if n > no, then the probability that aninfinite open cluster meets a given by a square is at least 1 - 10-1

Let n = no + 1, and let 5' be an by square in 22 . Suppose thatsome site x in S is in an infinite open cluster Then there is an infiniteopen path P starting at Let y be the last site on .P that is in S, andlet P' be the sub-path of P starting at in then P' is an open path from Sto infinity, using only bonds outside 5'. Let L i be the event that there isan infinite open path P' as above leaving S upwards, i.e with the initialsite y on the upper side of 5, the initial bond vertical, and all bondsoutside S. as in Figure 3 Let LO, L3 and be defined analogously,

Figure 3 An infinite open path P starting at a site a in a square Ssub-path P' [torn the last site y of P in S leaves S upwards.

rotating 8 though 90 degrees each time Thus P 1/2 (L 1 ) = P i/2 (L i ) knall i. We lime

Pip (LI U L9 UL3UL 4) > 1 - 10-1

As the L i are increasing events, it follows from H arris's Lemma by the'nth-loot trick' (equation (8) of Chapter 2) that P(L 1 ) > 1 - 1/10 foreach i: other wise, INV!) > 1/10 for each and, as the L ate decreasingand hence positively correlated, L?) 10-1, contradicting (5)

Recall that the planar dual of the square lattice 2 2 is the latticeZ2 + (1/2,1/2), and that we take a dual bond e* crossing a bond c of22 to be open if and only if e is closed Let 5" be an 11 - 1 by 11- 1

(5)

5.2 The Harris-Kesten Theorem revisited 127

Figure 4 Infinite open paths P, and P4 in the lattice V, leaving a equateS to the right and to the left The infinite open paths P. PI in the duallattice leave 8 1 upwards and downwards. If P( and may be connected byopen dual bonds in S' t hen there ate at least two infinite open clusters, onecontaining A, and one containing Pi

square in the dual lattice inside 8, as in Figure As the bonds of

the dual lattice ate also open independently with probability 1/2, and

as n — 1 = ne > no, the argument above shows that P i r(L;) > 9/10,where is the event that an infinite open dual path leaves the ith sideof Si . Thus the event E = fl L2 fl L43 1-1 LI illustrated in Figure I hasprobability at least 1 x (1 — 9/10), 6/10 > 0. The event E dependsonly on the states of bonds outside 8'. Thus, with positive probability,E holds and every dual bond in 5' is open But then the paths P. and.P:13 may be joined to form a doubly infinite path P' that separates theplane into two pieces. As P' consists of open dual edges, and an openedge of Z2 cannot cross an open dual edge, the open paths ./?, andlie on opposite sides of P', and thus in separate open clusters. Hence,there are at least two infinite open clusters

In short, starting horn the assumption IP, /2(1 1 ) = 1, we have shown

that with positive probability there me at least two infinite open clusters:

a contradiction It follows that P i pe(1 1 ) = 0, and hence that 0(1/2) = 0

El

128 Uniqueness of the infinite open cluster, and critical plobabilitics

Using Menshikov S l /WU/ it is veil, easy to complete l et another

proof of the fla t r is [Kristen Theorem. Them cm 13 of Chapter 3. restated

below

Theorem 6. Pot bond pe rcolation on the square lattice pm ! = pr = 1/2

Proof As misted in Chapter 4. Theorems 7 and 9 of that chapter. statedfor oriented site percolation. appl y also to tu t or iented bond percolation

and in particular. to bond percolation on E2 (Formalit y one can applythe theorems to the graph obtained front the line gr aph of 2:2 by replacing

each bond Ir y two ofmositdy oriented bonds ) In particular, lenshikoyisresult, 'Theorem 7 of Chapin 4. gives ',I li C.2; 2 1 = p r j (7,2 ) Since 0(1/2)

implies ph (;Y,') > 1/2 it tints suffices to prove that (2V) < 1/2

This is immediate flour Theorem 9 of Chapter -I and Lemma 1 ofChapter :3 Indeed. suppose that p.11(272 ) > 1/2 Then. In the first of

these results. as 1/2 is less than the critical probabilit y, there is an o > (1

such that LP iti(f 21() < exp(—mt) for all sites ,r and integers n. where

Tr 21 1 denotes the own that .r is joined by an open path to some siteat graph distance n front r Taking n large enough. we have

113 1/2 (ft "H----JJ) C 1/(10(M)

Let S be an n by n square in 2T2 As before. let 1-1(8) be the event that

thane is an open horizontal crossing of S II 1-1(S) holds. then one of then sites on the left of 5 is joined by art open path ill S to a site on the

right of S. tit distance at least n — I Hence.

(H(S)) < nP 1/2 <1/100

But this contradicts the basic fact that P 1 1 ,01(S)) > 1/2. which v,e

know from Corollary 3 of Lemma 1 of Chapter 3

As the Harris- Rester/ Them cm is so fundamental. let us briefly sum-marize the different approaches to its proof that we have presented here.All the pools start hum the basic fact that either a rectangle has anopen Ina izontal crossing. or its dual has an open vertical crossing Then,to prose that p ill (Z2 ) > I/2, one may prove a Russo-Seymour-Welsh(RS \V) type theorem as in Chapter 3. 'elating crossings of rectangleso crossings of squares Alternatively. one can deduce the result horn

the Aizeurnan- Kristen- NtiNV/IMIt ' filet)/ C/11, Theorem 4 lb prove that

iii(G2 ) C 1/2, haying moved an RS NV type theorem. one can apply

one of tat ions sharp-threshold results as in Chapter 3 Alternatively

5 8 Site percolation on the Niangidat and square lattices 129

one can deduce that ph(Z-) = (Z2 ) > 1/2 directly Flom Monshikov'sTheorem (Fm the last pint: we do not need exponential decay of theradius as used above: the almost exponential decay given directly byTheorem 6 of Chapter 4 is more than enough ) The approach used inChapter 3 is perhaps mote down-to-earth, and simpler it/ any one givencase The advantage of the Afenshikow Aizennnt- Kest:en-Newman ap-proach illustrated in this chapter is that the tools rue very general, sogenetalizing the moth to other settings is easier. Of course. one stillneeds a suitable star ting point, given by sortie kind of self-duality

Let us note that, in this latter approach, moving that percolationdoes occur undo suitable conditions. which was historically much thel uodei part of the Flattis-liesten result, is vent easy: the deductionhour Alenshikov's Them em is very simple. and can be applied in a peat:variet y of settings. In contrast, showing that percolation does not occur.which was histoticallt the easier part of the result, is more difficult: thededuction Man the Aizen n ian Kesten Newman Them ent is not quite sosimple, and requi t es write assumptions. We shall see this phenomenonagain when we consider percolation on (whet lattices

5.3 Site percolation on the triangular and square lattices

We next topside! the (equilateral) tit iangulat lattice T C IF:2 Pot defi-niteness, let US imam and scaler so that (0.0) and (1, 0) ale sites of T.and all bowls hate length 1 Portions of T me illustrated in Figures 5and 7 below Out aim is to show that psil (1) = = 1/2. As astinting point, we need a suitable sell-dualit y property

In bond percolation on V. the outer boundary of a finite open clustercan be viewed as tin open cycle in the dual lattice V ± (1/2,1/2) For

site percolation Oil T. it is east to see that any finite open cluster isbounded by a closed cycle in the 5anie lattice T. Also, an open pathin T cannot stair inside and end outside a closed cycle in T: indeed,the latter statement holds for site percolation on any plane graph, as acycle in a plane graph separates the plane into two components Theseobservations give a sufficient still ting point to enable us to prove thatMr(T) = p)(T) = 1/2, using the results of Menshiko y and of Aizenman,Kesten mid Newman. In fact, as in Chapter 3, it is easy to prove astriking ; huge-scale' consequence of the self-duality As usual, we writeP„ t he percolation nwastue undet considelat ion, in this case, for P)1/

Lemma 7. Let 1?„ he the rhombus in until 11 sites mi a ,side s"loam


in Figure 5, and let fl(R,,) be the event that there is an open path in Tconsisting of sites in R.„, starting at a site on the left-hand side of R„,and ending at a. site on the right-hand side. 'Then P 1/2 (11(1?„)) = 1/2

for every rr >

• V•• • •

.A. ••• • •Figure 5 to rhombus nu: solid citcles represent open sites, and hollow circlesclosed sit

Plug. Let IiI (R„) be the event that there is a closed path in R„ joiningthe top of R,, to the bottom. Reflecting R„ in its long diagonal, andexchanging closed and open, we see drat

1E„(11(R„)) = Pr_glit'(/?„))

for any p and any n. In particular, Pit,(H(R„)) = 1P it (17 *(R„)). Itthus suffices to prove that

17[72(H(R„)) +FP1/2(r(R„)) = 1

As in Chapter 3, we shall prove the much more detailed result that,

whatever the states of t he sites in R„ exactly one of the events 11(1?„)and 1 7 *(1?„) holds..

The proof is essentially the same as that of Lemma 1 of Chapter 3,although the picture is somewhat simpler One can replace each site ofT with a regular hexagon to obtain a tiling of the plane. Thus, what wehave to show is that in the game of Hex, no draw is possible: if all ti rehexagons corresponding to R„ ate coloured black m white, then eitherthere is a black path from left to right, of a white path from top tobottom, but not both (On a symmetric board, it follows easily that thefirst player has a winning strategy. )

5 .9 Site percolation on the triangular and square lattices 131

To see this, we shall consider face percolation on the hexagonal lattice,which, as noted in Chapter 1, is equivalent: to site percolation on T Moreprecisely, let; us replace each open site of R„ by a black hexagon, andeach closed site of R„ by a white hexagon, and consider additional blackhexagons to the left and right of R„, and white hexagons above andbelow R„, as in Figure 6

Figure 6. A partial tiling of the plane corresponding to Figure 5, obtained byreplacing each open site in Ro by a black hexagon and each closed site by awhite hexagon, with additional black and white hexagons around the outsideThe thick line is a path separating black and white hexagons, starting at :r,with black hexagons on the right This path must end at y (as shown) or at re

The rest of the proof is exactly as for Lennart 1 of Chapter 3: let Ibe the interface graph formed by those edges of hexagons that separatea black region horn a white region, with the endpoints of these edges asthe vertices. Then every vertex of I has degree exactly 2, except for thefour vertices ,r, y, z and w of degree 1 shown in Figure 6 The componentof I containing a is thus a path. Following the path star ting at thereis always a black hexagon on the right and a white one on the left, sothe path ends either at y or at ro In the for men case, the black hexagonson the right contain a path iu T witnessing H(R„), in the latter case,the white hexagons on the left witness V"(.11.,,) As before, 11- (1?„) andV"(R.„) cannot both hold as other wise Kr, could be drawn in the plane

Using the results of Menslrikov and of Aizeinn, Kesten and Newman,it is easy to deduce that the critical probability for site percolation on T

is 1/2, a result due to Kesten [19821 Menshikov's Theorem (Theorem 7

132 Unpteney; of the infinite open Haslet and et Hirai ptvbabilitics

of Chapter -1) cells us that m = p H in this context: horn now on, wewine Th. for their common yable

Theorem 8. Let I be the equilateral trianottla, lattice in the pla ns

Then ms.(T) = 1/2

Proof By Theorem 7 of Chapter 4 we have M i (T) =

Suppose first that /VT). it; (T) > 1/2 Then, by Theorem 9 of Chap-ter 1, we have exponential decay of the radius of an open cluster atp = 1/2, i e., there is an a > 0 stud/ that 111 1/ 9(0 < exp(—or/).. Defin-ing R„ as in Lemma 7, any of the sites on the right-hand side of isat distance at least it — t hour any of the a sites on the left-hand side.so

Pit2 (11(R„)) < Itifi l/2 (0 :) n exp(-0(n — 1 ))

As n — oc the thud humid tends to zero, contradicting Le11/1118 7Suppose next that p;'(T) = M i (T) < 1/2, so 8(1/2) > 0 Then. M-

I heorem 4, in the p = 1/2 site percolation on T there is with probabilityI a unique infinite open cluster Let /1„ be the hexagon centred at theor igin h " sit es on "eh side dm)" iu Figure 7 As U„ = T, if

n is large enough then the ihr-p t obability that some site in H„ is inan infinite open cluster is at least 1 — sap. Numbering the sixsides of 11„ in cyclic order let L 1 be the event that an infinite openpath leaves horn side i Mote precisely, L i is the event that thereis an infinite open path in T with initial site on the nth side of fl„

(we may include both cot nets), and all other sites outside ll„. Thenj i L 1 is exact') the event that there is an infinite open cluster meeting

„ > ks the events 1, 1 me increasing. and,H 1 — 10<6by symmetry. each has the sauce probability, it follows hum Iiattis'sLemma (Lemma 3 of Chapter 2) that P i /,(L) = > 1 — 1/10for each I

Let L is be the event that an infinite closed path leaves ./I„ from theitir side. Then F„ (L;) Ifil_.„(Li), so

E")/0(Li) = > 1 — 1/10

Hence, with probability at. least 1 — 4/10 = 6/10 > 0, the event E =

fl n fl L .; holds: this event is illustrated in Figure 7. Now E

is independent of the states of the sites in 11„_ i Thus, with positiveE holds and every site in /1„_ 1 is closed But then the

closed paths R„`„ R guaranteed by the events L and L may be joined

u.8 .5Vic percolation on the Niangalar nil squat( lattices [33

Figure 7. The hexagon Ho with the initial segments of infinite open paths/41 and P4 leaving its 1st and 4th sides and of infinite closed paths P; Pr;leaving the 2nd and 5th sides \Miaowr the states of the gte t sites (and theandrawn sites), the event. E = L, n n fl L; holds

to forth a doubl y infinit e closed patil sepal. at ing t he open paths PI and

Pi guaranteed by L I and L 4 It follows that. with positive inobabilih.Untie ace at least two infinite open clusters, cunt-Indicting Thement

An alternative proof of Theme/a 8 is given in Bollohas and Montan12006H, based on an RSW type theorem and the sharp-threshold tesultsin Chapter 2

The method used to prove Therrien' 8 above may be applied to sitepercolation in the square lattice This time, the claim} probabilitycannot be obtained in this way, as the lattice is not self-dual. Indeed,let A D =Z2 be the plaint] square lattice, and let Az be the graph withvertex set Lt. in which any two yet Hefts at Euclidean distance 1 or J/I?7are adjacent Thus Az is obtained from Ao by adding both diagonalsto each face of AD It is easy to see that a finite open cluster in the sitepercolation on Ao is bounded by a set of closed sites that form a pathin A. and vice versa Also, a path in Ao cannot cross a path in Azwithout the two sliming a vertex 'These observations ate the starting

134 Ull'iglieliGSS of the infinite open cluster and critical probabilities

point for tire proof of the 'duality' result for Ao and AE, Theorem 10below.

Let PI, be the product probability measure in which each site ofopen with probability p. and let I3 be a rectangle in 7L2 For A A D orA = A D , let HA(11) be the event that there is an open A-path crossing 11

horizontally, i e a set of open sites of I? that forrn a path in the graphA crossing R horizontally Similarly, let VA (R) be the event that there isan open A-path crossing R vertically. The following result correspondsto Lemma 1 of Chapter 3

Lemma 9. Let A be one of A D and Az, let A* be the Whet. and let

R be a rectangle in Z2 11 7hatettei the states of the sites in R. there is

either an open A-path crossing I? from left toright, at a closed A"-path.

Glossing R from top to bottom. but not both In particular.

Pp (H.\ (R)) +P 1 4,,(14A . (R)) = 1. (6)

Lemma 9 says that, if we colour the squares of an a by n i chessboard black and white in an arbitrary manner, then either a rook canmove hour the left side to the right passing only over black squares.or a king can move from top to bottom using only white squares, butnot both. Bollobas and Riordan [2006b] gave a very simple proof ofthis result; this proof is eery similar to the corresponding argumentsfor bond percolation on Z2 and for site percolation on the triangularlattice presented here Figure 8 below, reproduced from Bollobas andRiordan [20061d, is essentially the complete proof Although this factis not needed for the proof, let us note that the tiling in the picture isa finite part of the lattice (I,8 2 ) shown in Figure 18. The same latticewas used in a different way in the proof of Lemma 1 of Chapter 3.

Theorem 10. The critical probabilities for site percolation on the lat-

tices A D and AD obey the relation (A0 ) +K(A D ) = 1.

This result was first proved by Russo [1981] (see also Russo [1982]), byadapting the original arguments for bond percolation on Z 2 , in partic-ular, the RSW Theorem An alternative presentation of this approachis giver/ in Bollobiis and Riorclan 12006bj Once again, Theorem 10 iseasy to deduce from the general results of Alenshikov and of Aizenman,Nester' and Newman; we shall describe briefly the steps in such a de-duction.

5.3 Site percolation o the triangular and square lattices 135

I//

Figure 8 A rectangle R iuE 2 with each site drawn as au octagon, with anadditional row/column of sites on each side 'Black' (shaded) octagons areopen. Either there is a black path from left to right, or a white path (whichmay use the squares) from top to bottom Following the interface betweenblack and white regions starting at re, one emerges either at y or at w In thefirst. case (shown) t he event. K ALI (B) holds Otherwise V(1?) holds

Proof Once again, by kfenshikov's Theorem.. we have MI =p-1. for 10and for A9, so it is legitimate to write p for their connnon value

Given an assignment of states to the sites of by a A-clusler wemean a maximal connected open subgraph of A, where A = A 0 or Az.

Suppose first that ms (A iD) pr,(11.0 ) > 1 Then we um„y choose ap E (0,1) with 1 — pis,(A9 ) < p < p.,1(A0 ) Note that 1 — p < ps,(110)Taking each site of Z 2 to be open independently with probability p,by Theorem 0 of Chapter 4 we have exponential decay of the radius ofthe open A 0-cluster containing the origin, and exponential decay of theradius of the closed /1 0-cluster containing the origin For large enougha, this contradicts Lemma 9 applied to an rr by n square.

Suppose next that pr.(A0 ) p;1(A9 ) < 1 Then there is a p withp > p,(A0 ) and 1 — p > Ths,(Az) Taking each site open independentlywith this probability, by Theorem 4 there is, with probability 1, a uniqueinfinite open /10-cluster, and a unique infinite closed Az-cluster. Itfollows as before that with positive probability there are infinite openAD-paths leaving a large square S from two opposite sides, and infi-nite closed As-paths leaving S from the remaining sides. Hence, with

130 Unaptencss o/ the infinile opal rin g let awl (chiral probabilitics

positive mobability them ate at least two infinite open Any-clusters a

cont //diction q

As we shall see in the next section. Theorems S and 10 ate special cases

of a mote general result (Themern 13) concerning symmettic lattices

5A Bond percolation on a lattice and its dual

The results of Nlenshikov and of Aizemnan. Kesten and Newman imply

that. under a mild symmetry assumption. the e t hical probabilities lot

bond pe t colation CM a planar lattice A //rid on its platun dual A' satisfy

11? (A) + 11: (A') = 1 (7)

When A Z2 , the /elation above is exactly the Hattie Resten heo-

tent Later, we shall prove (7) in sonic generalit y (Theorem 13); first,

we illustrate it with another simple example As before. let I be the

(equilatet al) triangular lattice in the plane. Let /I be the planin dual of

defined in the usual way Taking the sites of H to be the centres of

the laces of T, then H is the (tegulat ) hexagonal lattice, or honeycomb:

see Figure 9..

Elgin e 9 Portions of the iangulat lattice T and its dual 11, the hexagonal

of hancyromb lattice

As we shall see later. the critical probabilities for bond pet colati on

4 Band percolation an a lattice and its dual 137

71 and on H have been deto mined exacth by NVienuan (1981d confirm-ing a conjectme of Sykes and Essatn 11963; 1961] For the moment:, weshow only that these critical probabilities stun to 1

Theorem 11, The triangalat lattice T and honeycomb lattice H satisfy

pc.1 ( T )± v!)(if) = 1 (8)

Proof The result follows easily front the general results of Menshikovand of Aizetunan. Is:esten and NeWinall. Suppose first that p"1.(11 ) -1-

1 4!(//) > 1 Then we may chose p E ((l,1) with p < .1)!!(T) and 1 — p <

p!)(11) Let us take the bonds c E E (T) to be open independently withprobability p, and each dual bond c' E E(H) to be open if and onl y if cis closed. Then la Menshikov's Theorem we have exponential decay ofthe t adius of open clusters both itr T and hi H.. Hence, taking a hugeenough 'rectangle' R as in Figure 9, with probability 99% t here is neitherau open path in 71 massing R how left to tight, not au open path in Hc t ossing I? limn top to bottom But by planar duality, them is alwaysa path of one of these two types: this is a special case of Le11/111i1 2 ofChaplet 3, w hose proof is the same as that of 1,0111111i1 t 01 that chapletIn this case. the figure obtained la replacing each degree d site of H otits dual by a 2d-gon, and each bond-dual bond pan by a squate, is the(3,12') lattice shown irr Figure 1$

To complete the moor of (8), it suffices to show that lot any p, at mostone of the petcolation probabilities 0(T; p) and 0(1-1; 1 — is strictlypositive 'This follows from Theorem -I as above: if both 0(1;p) and

— p) ate strictly positive then, taking bonds of T open indepen-dently with probability p, with probability 1 them is a unique infiniteopen cluster in T, and a unique infinite open cluster in II Considering ahuge enough hexagon in T. it follows as berate that with positive prob-ability thew ate infinite open paths in 71 leaving the hexagon flout the1st and Std sides, and infinite open paths in H leaving from the 2nd anddth sides. If the bonds of H inside the hexagon me also open, we find adoubly infinite open path iu H separating two infinite open componentsin It a cmaradict ion q

Having proved (7) in two special cases. fin A = V, and rot A = 21,we bun to a considerably more genet al result The arguments we havegiven so lar used the fact, that A had a suitable rotational symmetry,of onto 4 in the M i st case. and order 6 in the second In fact, theweaker assumption of onto 2 rotational symmetry is enough, although

138 Uniqueness of the infinile open (larder and critical probabilities

one has to work a little harder to obtain (7) in this case Also, there

is a natural generalization of (T) to certain settings in which bonds of

different 'types' may be open with different probabilities. In this context,

it is convenient to wad; with a weighted graph (A, p), i.e , a graph A

together with an assignment of a weight p c E (0,1] to every bond e of

A Fin each weighted graph there is a corresponding independent bondpercolation model NI ARA, p), in which the bonds of A are open

independently, and each bond e is open with probability pc

To state a formal result, by a planar lattice we mean a connected,

locally finite plane graph A (i e a planar graph with a given drawing'in the plane), with V(A) a discrete subset of 1R 2 , such that therm are

translations 21,, and of IR2 through two independent vectors t t i and

rto each of which acts on A as a graph isomorphism In particular. all

the Archimedean lattices are planar lattices Recall that two sites

and a', or two bonds c and e', are equivalent ill a graph A if there is

an automorphism y of A mapping r' to c to c' Note that anylattice is a finite-type graph, in the sense that there ale finitely many

equivalence classes of sites and of bonds under this 'elationTo allow for models in which edges have different probabilities of being

open.. we define a weighted planar lattice (A, p) as above: A is a planar

lattice, and there are two translations T„, and T. as above acting as

antonnophisnis of (A,p) as a weighted graph, i e preserving the edgeweights Perhaps the simplest non-trivial example is the square lattice,

with = p for every horizontal bond and p r = q lot ever y ye t Heal bond,

where 0 < p, q < 1 kesten [1982] showed that in this case, percolation

occurs it and only it p+ q > 1: see Themern 15 below Another simple

example is shown in Figure 10. Note that in a weighted planar lattice,

there can only be finitely many distinct edge weightsWe sus that a graph A drawn in the plane is centrally symmetric,

or simple symmeta lc, if the map a t--Y from P. 2 to itself acts on

A as a graph isomorphism For a weighted graph, this map shouldalso preserve the weights For example, the (weighted) planar lattice

shown in in Figure 10 is str ut/nettle if one takes the origin to be the

centre of an appropriate face If A is a planar lattice then, taking the

vertices of the planar dual A* to be the centroids of the faces of A,sav, one can draw A' as a planar lattice, as in Figure 10 We assume1:111°110/out that the bonds of both A and A* are drawn with piecewise

linear craves in the plane If A is symmetric, then we may draw A* so

that it is also symmetric The dual of a weighted planar lattice (A, p)

is the weighted planar lattice (A*,q) in which the dual e' of a bond e

5.4 Bond percolation on a lattice and its dual 139

Figure 10 The planar lattice A (solid lines and filled circles) obtained byadding diagonals to every fourth face of Z 2 If the horizontal. vertical anddiagonal bonds are assigned weights p, q, and r respectively, then A becomesa weighted planar lattice. The dual A' of A is drawn with hollow circles, atthe centroids of the faces of A, and dashed lines

has weight qc . = 1 — As shown by Bollobtis and Riordan (2006d1,percolation cannot occur simultaneously on a symmetr is planar weighted

lattice and on its dual

Theorem 12. Let (A. p) be a squattete ic weighted planar lattice. with

0 < p„ < 1 fat every bond c Then either 0(A: p) = 0 al OW:

where (A", q) is the dual weighted lattice

Proof Suppose lin a court adiction that 0(A; p) > 0 and 0(„V;q) > 0.In the proof of Theorem 4 it was not relevant that all bonds were open

with equal probability. Thus. writing Al and AP for the independentbond percolation models associated to (A, p) and to (A*, q), we see thatin each of Al and AI* there is a unique infinite open cluster with prob-

ability 1 As usual, we realize the bond percolation models 11I and Al*simultaneously on the same probability space, by taking the dual e* of

a bond e E A to be open if turd only if e is closed Throughout the proofwe write I? for the probability measure on this probability space

The basic idea of the proof is as follows: as before, any large squareS is very likely to meet the unique infinite open clusters in A and in A".If we had four-fold symmetry then, using the 'nth-root trick', we could

deduce that for each side of S. with high probability there are infiniteopen paths in A and ili A* leaving S from that side With only centralsymmetry, all we can conclude immediately is that there is some pair

110 Uniqucee s of the nrfiurlu open cluster and cr rhea/ probabililics

of opposite sides of 5 how which infinite open pat hs in A leave S with

high pr obalitlikn

The key idea is to move the 'corners' of S while keeping S the Sante

Mote precisely. iliStetuf of a square, We take S to he a circle whose

boundary is divided into four arcs A l, 24,/, and cot/sides infiniteopen paths leaving S hunt each A; If we move the dividing point;between two arcs, then pa ths leaving one become more likely, and paths

leaving tin other less Mich If we move the dividing point grachially, then

the ptobabilities will change in a roughly continuous umnner, so at somepoint they will be roughly equal 13v moving two opposite division points

while preserving symmetry. we can find a symmetr ic decomposition ofthe boundar y of 5' into four ales so that open paths of A leaving the

tom arcs a t e roughly equall y likely Now, using the lout th-rootfor even/ ate A 1 it is sett likel y that tittle is an infinite open path inA leaving S front this ate We cannot say that the same applies to ALas we have chosen the ales for A and not for A' We observe. however,that among out lour arcs there is sonic pair of opposite at cs of S hornwhich infinite open paths of A' leave with high probability Indeed. thisFollows from symmetr y and I he squaw-root hick This gives us infinite

pat hs in A. AL. A and A' leaving the aces of 5' in miler and, as before,we can deduce a contradiction lit showing that the two paths in A' maybe joined within .5. giving two infinite open clusters in A. We shall nowmake this aigument precise.

Let .5 = S r be the dicky centred at t he or igin with nadirs r Let E(S)

bet he event that some site of A inside S is in an initiate open clustet iu A,and let P(S) he t he men( that some site of A' inside .5' lies in an infinite

open cluster in A" \\Tilting D, for the disc bounded In 5, , we have

U, D, IR', so the union U , E(S, ) is simply the event that there isinfinite open cluster somewhere in A. and we have lin t , — NE(5,.)) =

and similar Iv for E' (5', ) Let S(A. p) he ;r positive constant that we

shall specify later Choosing r large enough, we have

NE(S,)) l—s 00d P(E' (S,))? I— (9)

For simplicit y, we shall assume throughout this ptoof that no site of A

ot A' lies on 5, and that tic bond of A ot A' is tangent to ,S, (Mote

pletiselv, recalling that bonds of A and A' me drawn as sequences of

line segments, we asSinne that none of these segments is tangent to S, )

This assumption is satisfied ha all but a countable set of values of r.

Lin the test of the wool we fix such an r large enough that (9) holds,

and mite tot S,

5 4 130nd petcolaijon on 0 lattice and its dual

141

Let e 1 1 < i < 4, he four distinct points on the boundfu y of S. 'nun-bered in anticlockwise order We write c for the quad/ uple (c,,/!,,c3We shall always choose these points so that no c; is on a bond of A of . A*We write .4 1 = A t (c) lur the boundary 01 r of S front cr to ciir t taking

e tr r If MC is a bond of A with o inside S and to outside. then ateleaves S front the arc A; iL travelling along the (piecewise linear) bond

ca! from V to 0', the last point of S lies on the ate .4 1 Let E 1 = E 1 (c) bethe event that there is an infinite open path in A leaping 51 from the arc

i.e an open path P = roo t co. with co inside S and outside S

for all j > such that non leaves S from the arc .4 1 ; see Figure 11

Figure I I Possible open paths Pt and Pa witnessing the events E t = (c)and Ea = E:dc) Usuallt, the bonds toe straight line segments, as in Pi butthey need not be

The precise details of the definition are not that important: the -soft'arguments we shall present go through with many minor variants Forexample, we could consider the last time the whole path leaves S. even ifthis is MI a hood soul with vo S Fot 'nice' drawings of 'nice' lattices,a bond typically crosses S at most once, so the condition is essentiallythat vo vf crosses Ai

Set

fdc) = INEi(c))

and

odc) — f dc) = P(Ei(c)"),

find define f and 91 similarly, using the dual lattice As an y infinite

142 Uniqueness of' the infinite open dusted' and critical probabilities

open path starting inside S must leave S somewhere, U 1 E1 (5') is exactlythe event E(S) The events Ea(S) are increasing, so then complements.E1 (S)" are decreasing Thus. by Harris's Lemma (Lennna 3 of Chap.ter 2), for any c we have

P(E(8)1

Front (9) it follows that

E;(c)c) P (Ei (c) c )= H (c)i=i er_r

gi(e) < 64

1.17( c ) EA

i=1

The key observation is that, as we move one point, (», say,the proba-bilities fi (c) changc, in a 'continuous' mannet . For a precise statement,it is mote convenient to work with m(c) The only properties of c thatthe event Ei (c) depends on are which bonds of A leave 5' from whicharcs A i .. Tints, as we move C. the probabilities Mc) and m(c) can onlychange when moves across a bond of A Of course, m(c) does jumpat these points Our claim is that there is a constant C = C(A,p) suchthat, at any such . jump, no m(c) increases or decreases by mole than afactor of C

Let c and c' = (e t , c!,.c3 , n i ) be such that exactly one bond c leaves5' front the ate c,cf, Without loss of generality we may suppose that

o)c1,,, ea lie in this order mound S; see Figtue 12 Thus, definingarcs .4 1 using the division points c, the bond e leaves S across the arcA I = eme2 while, defining arcs using the points c', the bond c leavesacross A{:, All other bonds leaving S do so across corresponding arcs

Thus, for i = 3, 41 the events E1 (c) and Ei (e) coincide, so

Re) = fe(ciLet E„ be the event that tire bond e is open The event E, (c) is defined

in terms of open paths leaving 5' across the arc ..4; If e is closed, then Itoopen path leaves S along the bond e, so which arc e crosses is in (Amami:Thus, the symineti ic difference of E i (c) and Ei (c') is contained in theevent E, In other words.

Ei (c) c fl = E;( c' )` n

5 4 Bond percolation oma a lattice and its dual 143

es CI

Figure 12. The effect of moving co slightl y to a new point c!): various bondsof A ate shown as dashed lines. We choose ct; so that a unique bond c leavesS between c2 and gt The ales .4; and AC are determined by A I = c i co.itt; = A2 = coca and Xi= etc:,

holds for each i Now E) (e)" and .6,1: are decreasing events, so, by Elan sLemma,

P(Ei(c)c n Etc,) P(E)(c)c)r(E))

Tints

me) P(E) (c')`) P(Ei (ct n

= iP(Ei (c)" n P(Ei(c)c)r(Ej cfn(c),

where

c = c(A,p) = int IP(Ect) = int — Pc) > 0,

as A has finite type and each Pc < 1. Similarly, gi (c) > cfn(c'), estab-lishing the claim

Set C = 11c > 1 Let us fix e l and e5 as opposite points of S Considermoving c2 from very close to c i to very close to c3 . At the star t of thisprocess, no bonds cross A t , so El (c) cannot hold, and (Li (c) 1 > go(c)Similarly, at the end, [Mc) = 1 > g i (c). Each time c2 crosses a bond,g i (c) decreases by at most a factor C. and 02 (c) increases by at most afactor C It follows that we may choose c,) so that

1/C G g i (c)/g2 (c) C

Let c. 1 be the opposite point to c2 Then by central symmetry we have

gi (c) = gi +2 (c) and g;. (c), 940 (c). (12)

111 Uniclacnctsof Un- open cluslcl and el Meal prababililks

Thus.

arh(c) = gi(C) 2 g2(e gi(C)4 /C2

Using (10). it follows that 9 1 (c) < C' 2E. and hence that

gi(e) < C:3/2613)

lot cow y i Front (11). there is some j with g(c) < 5 As (13) holdsfor even/ we Min' assume without loss of geneialitv that ) = L Thus,using (12) again.

q:1(c) = f/i(c) 5 s and Th(c) =512 (c) < 0 /2 E ( El)

It is now easy to complete the pool of Theo/ern 12, although, toavoid the need to consider exactly how the bonds of A and A" leaveS, especially Ilea/ the division points we shall introduce one moretechnicality

Let d be a constant (much) laigei than the length of airy bowl in Aor in A' Let F1 = li (c) he the event that there is an infinite open pathP in A leaving S across the ale such that no point of P lies withindistance d of any ci This event is illustrated in Figure 1:3 Let D(c) hethe event that all bonds passing within distance d of any c; are closedThew is a IllaXi1/1111.11 munhet of bonds of A that any disc of radius d canmeet:, so there is a constant c i(A, p) > 0 such that F(D(c)) >for any c Clear l y. if D(c) holds. then Ei (c) holds if and only if Fi(c)holds Using Hartis s Lemma as above, it follows that

h i (c) = N./2) (cl') Co i (c), (15)

where C I = 1/c 1 Replacing C I by 1/ minfc l (A. p), c t (A t . q)). thenboth (15) and the cot lesponding equation

11(c)= P(Fr(c) e ) 5 C f/7(c) (16)

hold fin any c

Let E = 1/(1.0C31'-CI ), noting that this quantityquantity depends on A and ponly, not 011 I 01 c From (14) and (15) we have

11 2 (0 (c) < (73/2 E < 1/10.

while fron t ( and (16) we haw

h;(c) IK:1 (c) < C 1 5 < 1/10

rid peicolaiiorilattice awl its dual 1-15

Figure 13. Open pat hs P, and P i witnessing the events F2(c) and F t (e) Theopen dual paths P( and PT witness i7(c) and ET (c) No site or bond of anyor these. paths wets the shaded clacks It follows that Pi, for example, leavesthe larger circle .5" through the arc col responding to As

Hence, with with probability at least I — 4/10 > ft.t he event

F = Fi (c) n F2 (e) fl Ps (c) fl P1(c)

holds, i.e., them are infinite open paths R and P i in A leaving 5' frontthe arcs and .4 4 , and infinite ()mut paths F,* and P.‘; in A* leaving S

from the arcs .4 1 and .4 3 , with no P1 or Rt passing with distance d ofthe endpoint of any arc 4 This event is illustrated in Figure 13.

-When F holds, there ate sub-paths P.; and P; of P, and P 1 leaving thelarger circle S' with radius] +d from the arcs corresponding to A, and.4 4 We may connect: .P1* and P3k by changing the states of all churl bondse" that meet 5' to open The corresponding bonds c lie entirely within5'. so after this change the paths p.!, and P.; are still open But then wehave, with positive probability, two infinite open paths separated by adoubly infinite open dual path This implies that there are two infiniteopen clusters in A, contradicting Theorem

As an immediate cotollat y of irheormn 12 we obtain the desired re-lationship between the critical probabilities km bond percolation on aplanar lattice A and on its dual, assuming only central svutineti

146 Uniqueness of the thfinitc open clustc; and critical probabilities

Them em 13. Let A he a symmetric piano] lattice, and let A" be itsAmu dual Then pc1.)(A)+ p lcl(A") = 1

Proof As before, the inequality plc,)(A) + plAA*) < 1 follows easilyhorn Menshikov's Theorem, by considering a large region in the planewhich must be crossed one way by an open path in A, or the other wayby an open path in A* In the other direction, pir!(A) +111. (A*) > 1 isimmediate from Theorem 12: if this inequality does not hold, then thereis a p E (0, 1) with p > pck.'(A) and 1 - p > (Al But then 0(A; p) andO(A'; 1 - p) are strictly positive, contradicting Theorem 12 q

Theorem 13 includes the Harris-Kesten Theorem, Theorem 13 ofChapter 3, and Theorem 11 as special cases. It also applies to manyother lattices, for example, all the Archimedean lattices shown in Fig-ure 18

Turning to site percolation, Kesten [1982] pointed out that the sitepercolation models on certain pairs of graphs are related in a way thatis analogous to the connection between bond percolation on a planargraph and its dual; he called such graphs matching pairs, and notedthat any planar lattice matches some graph To see this, let. A be aplanar lattice, and let A x be the graph on the same vertex set obtainedfrom A by adding all diagonals to all faces. For example, if A = A 0 , thenA'' = Ao The wool of Lemma 9 extends immediately to show that,for a suitably chosen 'rectangle' in A, whatever the states of the sites of1 1 (A) = 1,7 (A" ), either there is an open A-crossing from the left to thet ight. or a closed A x -crossing front the top to the bottom: to obtain thepicture corresponding to Figure 8, replace each site v of A with degreed by a 2d-gon that is black if v is open mid white if v is closed, and eachf-sided face of A by a white f-gon

If A is symmetric, then trivial modifications of the proof of Them ern 12show that 01,(A) and 0 1 _ 0 (A X ) cannot both be strictly positive, whilep-,5,(A) + ps,„(A ) < 1 is again immediate from 'A lenshiltov's Theorem,giving the following analogue of Theorem 13.

Theorem 14. Let A be a symmetric planar lattice. and let A' he the

graph obtained from A by adding all diagonals to all faces of A. Then

K.(A) + p,s,(A x ) = 1 q

As noted above, = AE , so Theorem 14 implies Theorem 10 Since ev-ery face of the tr hingular lattice T is a triangle, T X = T. so Theorem 1.1implies Them ern 8 as well

5.4 Bond percolation On a lattice and ids dual 147

We conclude this section with an application of Theorem 12 to aweighted graph. Let (2 2 , p,., pr ) be the graph 22 , in which each hot izon-tal bond has weight p, and each vet deal bond weight pm Kesten [1982,R 82] showed that the 'cr itical line' for this model is given by px ±pg = 1.

Theorem 15. Let 0(Th„ p„) denote llw probability that the origin is in aninfinite open cluster in the independent bond percolation on (22 ,p,, p„)

Fm 0 < p,,, pr < 1, we have 0(p„,,py )> 0 if and only if > 1

Proof The planar dual of t he weighted graph A = (2 2 , p„, p„) is A' =(22 + (1/2,1/2), 1-6, t -p„), the usual dual of 2 2 with weight 1 -py oneach horizontal bond, and weight 1- on each vertical bond Rotatingand translating, A' is isomorphic to (Z 2 ,1 - p,, 1 - pil).

Suppose first that O(p,,,p„) = 0 for sonic p„, p,, with p„, > 1,and fix 0 < < p„. and 0 < jig < pr with 11, > 1. By theweighted version of Menshikoy 's Theorem, the radius of the open clustercontaining a given site of A' = (22,[4,p'9) decays exponentially. As1 - p'y < Tit and I - < p'„, the same is tr ne in the dual, (22 +(1/2,1/2), 1 - 1 -1/J..) But then the probability that a large squarehas either an open horizontal crossing or an open thud vertical crossingtends to zero, contradicting Lemma 1 of Chapter :3

We have shown that the condition > 1 is sufficient for 0(p„,,p„)to be lion-zero To show that it is necessary, it suffices to show that

p, 1 - p) = 0 for every 0 < p < 1 Since (22 , p. 1 - p) is symmetric andself-dual, this follows from Theorem 12. q

As it happens, one does not need Them ern 12 to prove Theorem 13;the proof of the Harris-Kesten Theorem given in this chapter adaptsimmediately Indeed, suppose that O(p, 1 - p) > 0 Considering a largesquare S, it follows from the 'four th-root' trick that there is some sideof S from which an infinite open path leaves with high probability Ofcourse, the same holds for the opposite side The dual weighted lattice isisomorphic to the original lattice rotated through 90 degrees, so infiniteopen dual paths leave the remaining two sides of 5' with high probability,and one can complete the proof as before. In the next section we shallapply Theo/ em 12 to prove a mot e difficult result, that a cer talc analogueof Theorem 15 holds for the triangular lattice

118 Uniqueness of the infinite! open (A ide, and e l itical prohabiltlics

5.5 The star-delta transformation

Sykes and Essam 119631 noticed a second connection between bond pet-colation on the hexagonal and ttiangulat lattices H and T. other thanthat given by duality This connection involves the star-thavilln

formation 01 star-delta transformation, a basic transformation in thetheory O f electrical networks To describe this, let G I and Gi be thetwo graphs shown in Figure lt Suppose that the bonds of G I ate open

eit

y

Figure Id A triangle, G I and a star, 02, with the same :at lac:lune ' sitesand

independently with probability p i , and those of GO with p t obabilitn ya.In either glaph, there rue five possibilities fot which sites among 1,r, y, :1me connected to each other by open paths: all thtee may be connected,none 111M , he connected and some pair may be connected to each otherbut not to the third In other wo t ds, the par tition of fr. y, cl inducedIn tety subgiaph 01 0 1 Of of 02 is {{:r. {{,r}, f lb ca oneof the three pat titions isomorphic to {{,r, {c}} These cases have theprobabilities shown below:

pairs connected probability in C I probability in at

;ill

+ 3/4(1 — pi)

none (l — /03

— p2/ 3 ± 3/1 2( 1 — P2

Pr ( 1 — Pt )2

p4(1 — Ai)

Serendipitously, there is a solution to the duce equations suggested bythe table above, i e., to

Pi + 3/4(1 — p i ) = P2,

( 1 — / 3 = — p2F iIIp2 (1 p ) 2 . and (17)

Pi — P0 2 = P_( 1— P2)

Indeed, the last equation is satisfied whenever = 1-pi1—p i Substitutinginto either of the first two equations gives t he stare equation.

iii —$pi=1=(l.

5 5 The star-delta tionsjhrination 1-19

This equation has a unique solution in (0.1 ), nameh

pp = 2 sin(7/18) = 0 3172

Let a, i 1,2, be the random (open) subigraph of 65 obtained byselecting each bond independently with probability pi . whew p i = pc,and pc = 1 – pa As all Hum equations in (17) me satisfied, the randomgraphs 0 1 and a, are equivalent with respect to the sites 11, and z:

We may couple 0 1 and a so that exactly the same pairs of sites front{ Ti m ate connected in 0 as in 0

In the context of independent bond percolation, each bond of a (usu-ally infinite) graph is open independently with a certain probability,which we may think of as a weight The obser ations above mean thatit a (finite ca infinite) weighted graph A has G as a subgraph, with bondweights pu, then We may replace G I by 6'0, with weights 1–pp. Ignoringthe Internal' site of this operation does not change the distributionof open clusters This is a simple example of the 'substitution method'that we shall return to in the next chapter

Using the star-triangle transformation, it is easy to deduce bow The-OFC111 11 that p(1'(T) = pp This result was derived by Sykes and Es-san [1963; 1961] without rigorous moof. \Vicuna' [1981] gave the firstrigorous moot, based on the star-triangle transformation and a Russo-Seyniour-AVelsh type theorem

Theorem 16. Let T be the triangular lattice in the plane.and FI the

hexagonal oi honeycomb lattice Then

p(1.'(T)= 2 sin(–/18)

and

p{IVI) = 1 – 2sin(rr/18)

Proof As before. it follows nom Nlenshikov's Themem that the twocritical probabilities associated to each lattice are equal, so it is legiti-mate to write pr:' for their common value.

Let H' he the graph obtained by replacing every second triangle Tby a star with the same attachment sites; see Figure 15 Then H' is iso-morphic to /1; we shall keep the notation separate to indicate the differ-ent relationships to 71, reserving H lot the planar dual of T. Infonnidly,bond percolation on I with parameter po is equivalent to bond perco-lation on H' with parameter 1 – po, so both ate supercritical. both me

150 Unigaeness of Ike infinite open. cluslet and critical probabilities

WAWA WAresWA' aA SSWA TA TataFigure 15 The triangular lattice 7, and the graph H' 01) red by replacingeach downward pointing face of 7' by a star. The sites o are the circles(solid and hollow); its bonds ate the dashed lines.

subevitic 01 both are critical By Theorem 11, it Follows that both are

critical, giving the resultMore formally, by a (10M(1411 in or in H', we shall mean one of the

triangles to which we applied the star-triangle transformation, or theresulting star in .11 ' Consider the probability measures 111,1 mi in whichbonds of T me open independently with probability pa, and nu,

CSN in which bonds of H' are open independentl y with probability 1 — Po.We have shown above that the restrictions of these measures to a singledomain D may be coupled so that the same attachment sites of D arejoined by open paths within D in the two measures We may extend thecoupling to all domains simultaneously by independence. Any path inT . or, in H' between two sites of T may be split into a sequence of pathsP1 within domains D i , with the ends of Pi being attachment sites of Di.

It follows that, under our coupling, two sites y of T are joined by anopen path in T if and only if they are joined by an open path in H'.

Recalling that 0 is a site of T, let Co be the open cluster of T containing0, and let C') be the open cluster of H' containing 0 Under our coupling,we have (2() n V(T) = Co. As Cf, is a connected subgraph of H' andsites of V(H') \ V(T) me joined only to sites in V(T), which have degreethree, we have 'Cid < 41C1-, n 1 ,1 (T)I Thus

IC 10) 1 411C01

holds always in the given coupling, so

111. 0co l n) POC(d POCH �

5.5 The situ-delta leeinsfortnatimi 151

for every a Letting n — we see that 0(H';1 - po) = 0(T; po) Inother words, as H' is isomor phic to the hexagonal lattice H. we have

0(11; 1 -po) = 0(T; po) In proving Theorem 11, we showed that, for any

p, at most one of 0(11; 1 - p) and 0(Thp) can be strictly positive Thus,

0 ( 11 ; 1 — Po) = O CT ; = 0,

which gives p,i (T)> po and pcl.'(1.)> 1 - po Since p(1)(T)+ plc)(11) = 1by Theorem 11, it follows that; Al' (T) = po and 144.11) = 1 - po. q

Let us summarize what the results above, the Harr is-Kesten Theo-/ern. Theorem 8 and Theorem 16, tell us about the critical probabilitiesassociated to the three regular planar lattices.

Theorem 17. Poi the aware lattice V. art have

mi ./ (E2) = 1 /2.

fin the trianyalai lattice T.

K(T) = 1/2 wad 1 . (T) = 9 n(d/18),

and lot Mc he:rayon& of honeycomb lattice TT

ple'(H) = 1 - 2 sine,T/18)

In a sense, the summary above is a little misleading: for these threelattices, four of the six critical probabilities are known exactly. but thereare very few other natural lattices km which even one critical prohabilit\is known exactly

Tire observation of Sykes and Essar [1903 .1 concerning the star-deltatransformation is a little more general: let G I be a triangle in which thebonds have probabilities p„, ph and pc, of being open, and G, a star inwhich the corresponding (i.e., opposite) bonds have probabilities r,,,and ra of being open Then G i and Go are equivalent if and only if

1'1)0 = - r n k (1.8)

lot fi. j.1.1 = {a. 0, el

Pa pap,: + (1 - P)PbPc ± pa ( 1 POP p„ph(i — ) / a/ /

and

(1 - p)(1 -pb )(1 -pc ) = (1 - /„)(1 -10(1- le)*

l„(1-ts)(1 -t„)+(1-1„),/,(1---/J ±(1 -t„)(1-

152 Unignenciis of the infinite open elimle, and critical probabilities

The equations (18) ate satisfied by taking = 1 — for each i andthen froth the remaining equations reduce to

p„pem — pa — ye — pc + I = 0 (19)

This more general ,tai-triangle transformation was used by Sykes andEssarn 119041 to study percolation on a ti iangular lattice in which thestates of the bonds ate independent., but the probability that a bond isOpen depends on its orientation They derive (normigoirmsly ) equation(19) lot the critical surface' in this three-parameter model

Using Theorem 13 in place of [Theorem 11, the proof Of Theorem Ingiven above adapts immediately to this weighted model, to give the

Following result

Theorem 18. Let = fir . p„. yz ) be the weighted le iangulal lattice

in which bonds in the three directions have weights Th. IL, and fi,

sprelively when.: 0 < pr . p„. fi, 0 if and otilg

+ P y P P ;PO) z > 1 q

This result g as Lamed by Grim/nett [1999[.. using ideas of Kristen[1982; 1988] In the light of VIenslnkm is Thement, the hind pint is toshow that percolation does not ()Celli when ys p„ T p, — pr y„p, <

Kristen [1982] deduced this result Flout a theorem that he stated withoutproof A N'elSiOn of this liniment that is in litany was !note generalwas later proved by Gandolfi, Keane and Russo L19881, but then resultassumes synunetiv untie/ reflections in the coordinate axes, which thismodel does not have.

A star-delta tt ansfot mat ion 'elated to that discussed above is impor-tant in the them, of electrical networks. where it has a much longerp iston The operation on the graph is the same. but the weights (re-sistances) t i anslo t aiireleutiv. to satisfy the different notion of equiv-alence (that the ' espouse i . net cadent at each attachment vertex.to each input. i e. set of potentials at the attachment vertices. is thesame) 6n electrical networks. it hums out that even sla t is etpthalentto kt triangle. and vice versa: see Bollobris [1998. pp .13 -H]

Rental kablv. Al t man 119811 w as able to use the star—triangle tianslot-mat ion with unequal edge weights to obtain t he exact critical probability

5 5 111P, si tu-della Emusformation 153

Cot a certain lattice, where each bond is open with the same probability

Let ,5 1e be the square lattice with one diagonal added to every second

lace, shown (rotated) on the left of Figure 16, together with its dual D

Figure lb 1 he lattice S obtained bow the swami Ire t ice by ridding a diago-nal to ever} other face, shown on the left (solid circles and lines) toget hen witits dual D (dashed lines and hollow cir cies) Fru clarity is draw n sepalateINon the light

Let S' he the lattice shown on the Hi in Figure IT below, obtained Inuit

S by replacing each of the diagonal bonds b t a double bond Then SI-.

with tinily bond open with probabilit y p. is equivalent to 5' ' . with the

otiginal bonds of S t open with probnbilitq p, and the new bonds open

with probability =1 — (I — p)' /.1 (As usual. the states of different

bonds are independent ) Appl y ing the star—triangle t t anstot motion. one

obtains t he lattice I)' bu nted hour D by subdividing ce t lain howls: see

Figure 17

Figure IT certain bonds of 5'' arc replaced by double bonds, and thetliangle-star t t ansibi mat ion is applied as shown. the resulting lattice is I)%vit h certain bonds subdivided

Taking the undivided bonds to be open with probability q = I —

p. and the divided bonds with probability q = q = I — p'. we see


that D' is equivalent to D, with every bond open with probability q.The conditions for equivalence in the star-triangle transformation aresatisfied provided (19) holds with p„ = = p, Pc= p' This conditionreduces to

— p — 6p2 + 6p3 — ps(20)

Using these transformations and arguing as for the hexagonal and tri-angular lattices above,\Vierman deduces that

p!),(S÷ ) = 0.101518

a loot of (20).

An Architnedean Ionia is a tiling of the plane by regular polygonsiu which all vertices are equivalent, i e the autornorphism group ofthe tiling acts transitively on the vertices. The square, triangular andhexagonal lattices are all Archimedean, the lattice 5' and its dual are

not The complete set of Archimedean lattices is shown in Figure 18,The notation, which is that of Griinbaurn and Shep pard [19871, is self-explanatory: it gives US the orders of the faces when we go round a ver-tex At this point we have essentially exhausted the list of Archimedeanhaft:CS Pi exact critical probability is known; there are two furtherexamples that may be easily derived from those above. Let K be thekagona'r lattice. shown in Figure 18 Then K is the lice-graph of thehone ycomb II, so we have

gdIC), p(1,(H),---- 1 — 2 sin( 7/18)

Also, let K' be the (3, 12 2 ) or extended Kagoind lattice shown in Fig-ure 18. Then IC + is the line graph of the lattice H2 obtained by subdi-viding each bond of II exactly once As noted in Chapter 1, the relationfki (11(i), p!(H) I12 is immediate: an open bond in the subdivided graphis only 'useful' if its partner bond is also open. Thus, as noted by Slidingand Ziff [19991, among others,

fise(K), pcIVI2 ) = p lAH) 1/2 (1 — 2 sin(rT/18)) L/2 (21)

In the next chapter we shall review some of the upper and lowerbounds for the critical probabilities of Archimedean lattices.

•••••• •A••••••n•nn• n• 114SUOMIn•nn•nn• .41y

•An•nn•nn• ............ASq nu ):

Triangulm: (36) Hexagonal: (63)

tome: (3, 3,6) extended Kagon (I. 82)

5.5 The stun-delta hamsformahmt 155

(a,6, 12) (3, -1, 6, -1) 1 -1)

4•1n•n••►a•• a•

•••ally A A A A 111aw IS • • FaVa•r• A•sffa a•Al • rA A

Va•• • Alwarar A A ValrAwA•0110•

SAY VANS •AT • A ATAYAT VA

(3' .1- , 6 ) (3 6)

Figure 18 The 11 Archimedeatt lattices, i e filings of the plane with regularconvex polygons in which all vertices ate equivalent The notation for theunnamed lattices is that of Griinbainn and Shep pard (1987) 10 of the latticesme equivalent under 'oration and translation to then 'Milo' images The finallattice. (31 ,6), is not, and is shown in hot h forms

6

Estimating critical probabilities

In genet al. there is no hope of dele l mining the exact critical probabilities

/OA) and pl,;(A) for a general graph A. even if A is a planar Ian t ice

Neverthel ess, there a l e mane ways of proving rigorous bounds on these

critical probabilities In this chapter we shall describe sever al of these.

soil ing with the substitution inclhod, a special case of which we saw in

the previous chaPlel

6.1 The substitution method

Tu describe the substitution method, we shall use the fenninolugi, of

wcightcd graphs: all our graphs will haw a weight p, associated to each

bond e. with 0 < < t We shall consider independent bond percola-tion Olt a weighted graph (A. p), whew each bond e of A is open with

probability p, independent Iv of the ot het bonds. We often suppress t he

weights in the notation. A weighted graph (G. p) with a specified set A

of attachment sites generates a 'widow partition n of A: two sites in A

are in the save class of El if they rue joined ln an open path in G As in

the precious chapter. we SUN" that ( WO weighted graphs G I . C, with the

saute set .1 of attachment sites ate equivalent if the associated I andoin

partitions Il i II, have the same distribution. i e if the col lespondiugpet colat ion measures can be coupled so that IT = 11 2 always holds In

general. exact equivalence is too much to hope forLet us saw that a partition 7ir. , of A is counsel than "Th. and mite

71 .) > if anNr two sites of A that ale in the saute class in 7 1 are also

in the same class in79 In other words. r, is (ionise' than r i if and

onl y if Ti is a 'ohne/new of in this context. a coarser partition is

'better'. as it will correspond to more connections in the percolation

model: this is the reason for our notation 'We say that a weighted

G.1 The substitution method 157

graph CI, is shott ipo than Cr, and WI > il rte DIM couple theco l I esponding percolation measures so that II is alwa ys coarser than

H I In this case Flo stochastically dominates FT,. Note that C I and Ga

t ue equivalent it and onl n if each is stronger than the otherLet A, and Aa he two infinite weighted graphs Suppose that A l may

be decomposed into edge-disjoint domains Dr j . i = 1.2, . each having

a specified set A j of attachment sites We assume that each D I is a

suligmpli of Ac that (l i cit union is A t and that t. WO dOtliltillti 1118V meet

onl y in sites that are at tachment sites of both Typically the graphs Dume all isomorphic Suppose that Aa has a decomposition into domains

hew each Du has the same attachment sites as D i \\: e have

seen an example of such a decomposition ;heady. ill connection with the

sum-triangle transformation Indeed wi le/ i ing to Figure 15 of Chaplet 5.

we IWIV take A l to be the tiitungular lattice. A, the hexagonal lattice

fr. the domains D I j to he ever y second triangle in A l and each Du to

he the cot tesponding star in A . , Aiguing as in the pool of Theorem 16

of Chaptet 5. since am, path bout one domain to ;mottle:: must pass

Huough an attachment site. the onl y inopei It of D I ; that is teleiant

tot percolation on A i is the induced partition 01 the set TI; Hence. if

is equivalent to 171 j fin eve/ then petcolat ion occurs 011 A l if and(t h if it Deems on Am fin any fixed attachment site the percolation

probabilities O(A j : 3 ) ;o ld 0(Aa: .r) ate exactly equal

Similarly. if Du > tot every then 0(Aam ) > 0(A 1 : ./.): one

call couple the percolation ;leashes on A, and A, so that any pair of

at tachment sites that a t e joined in A i ale also joined ill A, Tins tact

allows its to let iVe i elat ionsltips between the critical probabilities of two

lattices: if the ethical mobahility of one is known then we can hound

the critical p l ohabilitn of the other This technique is known as Hie

substitution method. and is due to Vieunan 119901At frost sight. it is not cleat hots one can tell whether a given weighted

graph is stronger than another, but Hume is a simple algorithm An up-

set ill the partition lattice on a set A is simpl y a set U of partitions of

A such that whenever E U and 71 > 71. then ira E Ll It is not too

haul to show that > (7, if and onl y if. foi even up-set. we have

E > P(H, E U): in fact.. this is an eas y consequence of Halls

'Matching Theorem [1935] (see also Bollolnis (1995. p 77j) hus, the

condition CD > G' t is equivalent to a finite set of pol nomial inequalities

on the weights of the bowls Let its illustrate this with a simple CNillipie

giving hounds On the critical probability p lT(it ) = = p!"(10

whew It is the Anytime lattice shown on the light of Figure 1 the

158 Eqimatilly dim( probabilities

Figtue 1 A two-step transformation 11'0111 the hexagonal lattice /I to theliagomet lat t ice K: first subdivide the bonds of to obtain the lattice ifsshown on the left. Then apply the star-triangle transformation to the origi-nal sites of /1 (middle figure) The result is the kagotn6 lattice (right-handfigure).

notation of atiinbaum and Shephatd [19871 for Ai chimedean lattices,is the (3,6,3,6) lattice.

Ottavi [19791 noticed that the Kagoin6 lattice may be obtained hornthe honeycomb or hexagonal lattice H by fist subdividing every bond,and then applying the stat-thangle transformation to the (non edge-disjoint) stay s centred at the original sites of H see Figure 1 (A moteinimitite sequence of stair ' equivalent ' graphs was shown in Figure 6 ofChaptet 1.) Let Ss denote the sla t with attachment sites {,r, y, :di inwhich each bond has height s, and Tr the triangle on {:v,:y,s} in tvhicheach bond has weight t We have seen that 8, is equivalent to T, if andonly if t po and s 1 — po, where po = 2 sin(x/18) -We would liketo know lot which pails (.s, t) we have > Th and lot which pans wehave Ss <111.

Recall that there ate 5 partitions of {x, y, z}: one itt which all threeare connected (are in the same part), one in which none ate connected,and three in which exactly one pail is connected We shall 100 to theseas the partitions of type 3, 0 and 1 tespectively, so the type of a partitionis the number of connected pairs. Repeating the calculation in the lastsection of the previous chapter, the probability that Ss induces a givenpartition of type i is Rs), where

I3(s) = li(s) = — and fo( s ) = ( 1— 5) 3 + 3 '4 1iw/2

The cott espondmg ptobibihities tot T1 ate given ht- Mt). with

firr(t) = ± 3t2 (1-1). yr = 1(1 — 1) 2 and tto(t) -= (I —

6 1 The substitution method 159

There ate 10 up-sets in the par tition lattice on a Once-element set: two

are trivial: the empty up-set, and the up-set consisting of all partitions.Any other up-set must contain the type-3 partition, cannot contain thetype-0 partition, and may contain any subset of the three type-1 par-titions. Let us write for one of the non-trivial up-sets containing )partitions of type 1 In this symmetric setting, we have Ss > Tr if ando i ly if four inequalities hold: each uj must be at least as likely in thepartition induced by Ss as in that induced by Ti

In the partition induced by S. we have (NUJ ) = h(s) +1 i(s), whilein that induced by T, we have P(U1) = g(t) + !in (0, so Ss > Tt if andonly if

ni(s)-1- lids) g3 (1)± Rh (0

holds for ) = Similtuly, Tr > told only if the reverseinequalities hold. 01 course, if (I) holds for ) = 0 and for . j= 3 then italso holds for j = 1 and j = 2, so there ate only two conditions to vet ify

We know the critical probability tot H. the left-hand lattice in Fig-ure 1 Indeed, writing p„ for p H or p1 (which are equal fin any latticeby Menshikov's Theorem), from equation (21) of Chapter 5 we lime

p(1.)(/12 ) = pti!(H) 1 /2 = (1— 2 7sin(11.8)) 11 = so

say As shown in Figure 1. this weighted graph 11,, with bond weightss, has a partition into weighted sous S, Replacing each stat with atriangle T, we obtain the Kagontê lattice with bond weights!. It followsthat, if 0(119:s) > U and T, > then 0(K:1)> U As o(it,;s) > 0 finany s > so.. we thus ha‘ e

mi,)(K) < int : T, > for sonic s > so inl ft : Ss

p(1.)(k) > slug/ : 8,0 > Tel

Solving the simple polynomial inequalities (1) fort with s = so, this

method gives the bounds

U 51822 < pcl ."(K) < 0 5-1128

These inequalities and the proof we have just given are due to Wier-man [1990] In the same paper, lie obtains the stronger upper boundpl:(K) < 0.5:335 by considering a larger substitution - replacing theunion of two adjacent: stars in HO by the union of the co/responding tri-angles in IC In this case there are lour attachment sites, so the partition

1(i9 Estimating t tlical 1)101)001,1u s

lattice is now complicated 13% using huge, and lilt get substitutions.hot ten and bet k g bounds m i t\ l ie ()With/v(1 1-lowc) \ el. the calculationsquickly become Wiwi/Oita' ii C111110(1 0111 ill a 11111\ C \la \ Using yxtionsmethods of simplifying the eill(ititat and it sttl/stitittion \\ it h six at-tachment sites. \Vic/in/an 120031)] shat wined these 'mitt/cis considrilabl

Theorem 1, Let Ii ix the ha q Dm( o, (3.6.3.6) lattice Then

9 5299 < phK ) < 0 5291 q

Pet tuning to the stint-itriangle Panstointation. the condition >

is fin some PluP oses. anueeessaliin strong. Suppose that we Wee a\\ eight ed aph A l with as a subs' aph (joined out \ at the attachmentsites). mud we replace S, bt 1, to Obtain A9 Nle would like condition'son s and t that allow us 10 deduce that 00/C01/1110/1 is mo l e likel y in Althan in A.1 Mote wriciseh we should like conditions that cosine thatthe went {0 — that a particular site (the might) is in an infiniteopen (lusty ' . is at least as likel y in A l as ill A . ) lVe write Ss :7-- 7 / itthis holds Ion all /Wits of weighted graphs A.)) /elated in the winwe have described. Oita\ i 119791 round the set of pai l s (s. ) lie which5, 1– .Ej

To present this t esult let us Ws( decide the states of the bowls inE the set of all bonds of A I outside 5. Note that E is also the setof bonds ()I A . ) outside T, Eon each attachment site p E u there111/0 1/1 Wilt not be 1111 open path limn 0 to 1' in A I \ S s . and therenia \ 01 /100,' not he an infinite open (inst il ()I A i \ meeting

°diet "°° 111 111C ("1/18 {{) } and { } 11111% of m in not hold\ Chen the states of the bonds in E. the conditional probability

that — x depends owl \\ Melt of the events and —hold in A i \ 5. Indeed. tl — x in A t if awl only it thew ale sites el.

E dial are connected within .5 – with 0 — and x inA i \ Hew. e l it.) is allowed

Much of the time, this conditional wobabilit \ is 0 (it 1: it 0— v andu rot some e E z} then 0 — x even if all bonds in S„ meclosed Similarl y if 0 f r fia all 1' e f.r C }. on r f x for allthen 0 — x cannot hold. whatwei the states of the hoods in 5, Thisleaves only One() non-t thin! cases: we must have an attachment site disat, with — :0 and a/101w/ site tr. tin) with 11)101 the Hindsite, we roan 1000 0 — x. 01 0 71-^ c 7 x In the last case.

— x in A t if and only it ,r and it me connected in .5, II 0 —c andx. then 0 — x A l if and 0111V if one of and : is connected to

1 The substitution method inl

y Slunk)/Iv if d sc., then 0 — sic in A i if turd only if one of irf

and c is connected to :J . in S, The relevant events in S s have respective

Pwilmbililies (s) ± ( fa( ± 2 f (s) and .f:Ws) + 2 fi(s)lane shown that PA, ,X) is a weighted awlage of the quan-

tities 1. 0, 13 (s) + h(s), and nr(s) + 2[ 1 (s), where the weights depend

on A t \ S. Fut Him more s PA2 (0 sc) is a weighted average of 0,

y3 (t) -I- in (1). and g3 (t)-1- 2g i (f) with the sonic weights, determined by

\IT/A i \S, Tints Ssbolds mecisciv when the two inequalities

E(s) ± [ 1 (s) � 00= !EU) and /3 (s) + 2f ( ) 113(0 20 1 (1) (2)

hold. i . when (I) holds tot = I and j = 2 This is a much tveaket

condition than S s > Tt flui trivetse relation Tr S. holds pun Wed the

revel se inequalities to (2) hold

Ottayi showed that il s so is the (*initial probabilits for [1,, and I =

0 52803. then E >- so the sultglaph S s of run weighted gin/tit A may

be teplaced and the probabilit y that a given site is in an infinite

cluster ( 01 that a given pair of sites ale connected by an open path) will

not decrease It might seem that the inequalit y p(1 9 fi) < 0 52893 follows

easily: un101 I mutt els. this is not the case As the Wit III al at gement is so

close to working let us ex tunine it in detail, to see where it fails

One would hope that. if T .5.. then percolation is at least as likely in

a graph Bunted In gluing together copies of 1st at tlwit illlath/nem sites.as in the col responding graph obtained from copies of 5, However. the

relation 8- S s allows its to replace (me cop%of Tj lw a copy of S, in

an 'outside wank w [licit is the same beton . and after the substitution.

but it does not allow its to conti nue and !vitiate a second copy After

enlacing the first cop y. the outside gl aphs E l and Eg ate different: in

Eg Nu , have ahead -n replaced a copy of S s by 7r, One might hope that.

as IT, is 'bet lei' than E l them is no teal problem But what does'better' mean? ft could mean that an y connection in the outside graph

E i between an at tachment site of mu second substitution and ft or isc

is also present in Es Until we have looked at the second substitution.we do not know which connections we will tric l inic., so we should impose

the condition that the first substitution preset yes all such connections

(while pet Imps adding new ones) ilhe condition Tr S. does not allow

us to do this only to p i esti ' NC a connection chosen in advance

The arguments above illustrate the power of the notion of stochastic

domination: if I) > S s then it is ye t y cans v to prove that we nun replace

as main copies of S s by copiris of Tt as we like I In key to the application

162 Estimating critical probabilities

of the substitution method is to find suitable weighted graphs an

with Gr > ay.\Viet/Ilan 12002] used the substitution method to obtain bounds on

piti(A) for other Archimedean lattices A, obtaining the following result.

Theorem 2, Antony the Atchimedean lattices A, the extended I< againá,

07 (3.12 2 ). lattice 0013i70/2CS p!?(A), with

0.7385 < pciVii+ ) < 0 7-119

Although we have described the substitution method only lot bondpercolation, essentially the same method can be used to studs site per-

colation. For example, \Vietnm [1095[ adapted Ids method to obtain

an upper bound on p-cs,(7.9)

Theorem 3. The et itical probability pi,(22 ) for site pet ()lotion on the

S01107V lattice satisfies pis.(22 ) < 0 679+192

Po" a list of other rigorous bounds on the ccritical probabilities for

lattices. see Wiennan and Pm viainen [2003]

6,2 Comparison with dependent percolation

Another method of bounding critical probabilities is implicit in the use

of dependent percolation in Chapter 3. Let us Wrist] ate thus by giving

an upper bound fill icii(Z 2 ) = M i (22 ) = 14(22 ); we shall mention a

much more sophistica.ted version' of this idea later. As usual, we write

= P_:. t,for the independent site percolation measure on 2 2 , where

each site is open with probability p; we denote this model by M. Let

be a parameter to be chosen later, and partition the yet tex set of Z2

into C by ( squares S,„. e E Z2 Thus, ha v = (a b),

S„ = = Ur, y) E 2,72 : at 5 < (a +1)1', Itt < y < (b +

The set of squares St has the structure of Z2 in a natural sense To make

use of this, for each bond e = u e of Z2 , let R = S„ US,.. so let is a 2t by

(or t by 20 rectangle. If e and are vertex-disjoint, then the rectangles

R.„ and R./. are vertex-disjoint Thus, if we define a bond percolation

model M on 22 in which the state of a bond e E E(22 ) is determined

1w the states of the sites in Rt . then the associated probability measure

on 2 E(:2) will be 1-independent. The idea is to define IC/ so that

6.2 Comparison with dependent percolation 163

an infinite path in Al guarantees an infinite path in tare original sitepercolation Al

We have seen a way of doing this in Chapter 3, using 3 by 1 rectanglesfor clarity in the figures We use the same idea here Recall that 11(R)

denotes the events that a given rectangle R is crossed horizontally by anopen path, and 17 (R) the event that I? is crossed vertically by an openpath. For a horizontal bond e = ((a, b). (a + 1, b)) of Z2 . let (AR() bethe event Win b)) illustrated in Figure 2. For a vet tical bond

Figure 2 The figure on the left shows open paths guaranteWng that t he event0(10 holds, where c = ((a. b), (a + l.b)) is a horizontal bond of 2:2 ; thesites shown are open 'The corresponding event lot a vet deal bond c is shownschematically on the right.

= ((a, 6), (0, b + 1)) O r 22 , let 0(11%) = 1 .7 (R,.) n H(SR„ 0 ) in either

case, let e be open in Al d and only if 0(RJ holds Since hor izontal andvertical crossings of the same square must meet, if there is an infiniteopen path in A/ then there is an infinite open cluster in Al Note thatthe probability Pp (G(Rt .)) is the same for all bonds c

Let p i = p i (Z2) be the infimum of the set of p such that, in any 1-independent bond percolation measure on 7G2 in which each bond is openwith probability at least p, the origin is in an infinite open cluster withpositive probability. As we saw in Chapter 3, it is very easy to showthat p i < 1 Here, the value of p i is important; we shall use the resultof Balistet Bollobris and \Valte i s [2005] that a t < (1 8639; see Lemma 15

of Chapter 3Suppose that for some parameters s and p we can show that

lt(P)=Pp(Cl(R.,)) >

Then lTr is a 1-independent measure on 72 2 in which each bond is open

161 Estimating . 'cal probabilities

with some probabilit y p > so with positive Limitability there is au

infinite open path in Al and hence in II/ 'Thus p> p11(Z11)

Suppose that p > p11(Z2 ) is fixed Then, by 'Theorem 10 of Chapter 5,

we have 1 — p < t ,11(Am), where A D is the square lattice with both

diagonals added to every face Exponential decay of closed clustets

A D follows by Menshiliovs Trhecneum so the probability that a 2( by t

rectangle R is crossed the shott way by a closed A D-path tends to zero

as ( Hence, by Lemma 9 of Chapter 5, the probability that R

is crossed the long YVilV by au open path in An tends to 1 It follows

that i f (p)--(• 1, so thew is sonic 1 Stith f hat l i (p)> CI 8639 > Thus,

in plinciple. Hie method above gives arbittarilr good upper bounds: foreach (', find the minimal rot an almost mini/nap value of p, of p such

that heir) > 0 8639. Then each p, is an tippet bound on p11(2, 2 )„ and

the sequence p, converges to pi1(Z 2 ) boo, above. ft is easy to check that,

',f p ) = 6p5 M 1 16p6 q2 +8p7 q 1ps , \\dune q = 1 — p, giving the bourn!

p11(Z) < 0 8798 With a computer, it is eas y to show that

fa(p) 117/tq l ° ±1399p9 q91737p lugs-1027p 11 q 7 = 5-166p12q°

+ 1527p 1 "q 1.' 2335/ J il t/ I - I- 7571 , 15 q3153/) 16 ,12 -F 18p ri+ pis

giving p1(21: 2 ) < 0.815 An exact (ruination of 1,(p) gives /.)`,(II32) <

0 817 Linlin unat el n t he st r night il/Wald met hod of evaluating rpr

mnd count inglia which of t he 2' 2 configut at ions in a 2l In 1 rectangle

11), the event 0(11, 1 holds, quickh becomes imm act lent. at mound I = 5

\\Ten the subst it Mimi method can be applied. it tends to give bet-on bounds for the same computational shirt Note, hotveier. Hurt

the method described hew is much more robust Fin example. fi l eg-

1 1 1e/Iiii( IS in die graph Me riot a ploblem plodded we can show hat

12'1,(G(1, )") > i > pr ha every bond c one can perform different

substitutions in diffetent pails of the gtaph the substitutions lune to

fit together exact) to give the stt ucHne of a graph with known critical

probability This will often not be possible

Another very imputeant adiant age of the 1-independent approach is

that, if one is p l epa l ed to accept an ewer probabilit y of 1 in a million.

say. or 1 in a billion. then good bounds can easil y he obtained by this

method Indeed. there is a yeti- easy way to estimate the numerical

value of E(p) very plecisek: generate configtuations in a 2( by

rectangle I?, at random using the ['wasn't ,and count the number

.I/ of configurations 1hr which 0(1?,) holds. The random number Al has

a binomial dishHa t tion with parameters A . and p) so if A is huge.

n',2 Commi t Cion with dependent percolation 165

then with lel \ high probability 11//A will he close to /r(p) The y IreVpoint is that one can boned the error probabilit y. ercu without knowing

ir (PrGiven any > 0, one Call give a simple procedure for producing upper.

bounds on /OE') ry ltich provably has probability' at most E 0[ givingan incorrect bound: foist (by trial and 0110 1 . or guesswork) decide onparameters I Will p for which )r(p) seems likely to he huge enoughThen calculate numbers N and Ma such that TdA. > J1,(0 ) < whew

Bi(N. 0 8639) Then generate A samples as ;drove. m i d if Al > 1110of t hese have the propert y Ci(g). assert pe r r....- 1 p as a bound IL

in fact 0,(23 2 ) > p. then h(p) < 0 8639. and t he probability that the

sampling procedure generates at least .3/0 successful trials is at most

This method works ver y well in 0010 and the dependence ()I theI mining time on is ver y modest For eXa/111A0. using the parameters

= 10 000. p = 0 591, N = 1000 and .110 = 935 with a moderatecomputational efhat we obtain the hound i t,".( 2 ) < 0 591 with all 19 lotplObabilit \ Of E < 10-12

Similar Iv, 161 the matching lattice AE with vet I CX set out bonds

between each pair of vertices at Euclidean distance I or v' 71. using the

pat ;mulcts l = 20 000. p 408. N = 1000 and A' = 935 we obtain

K( A3) < -1(18 w" a c""lidellue of1 1g-12 Together these boundsgive

/):1( ) E 10 592.0 5911

with extremely- high confidence. A little 'note computational effort (1=80.(100. N = 1000. Itltt = 915) gives the result K(Z 2 ) E [(1.592 0593]with 99 9999cA confidence

Note that we cannot: sat- that the probability that pr(r 2 ) lies in theinlet val (0.592,0 593] is at least 99 9999%: the critical probability is nota random < / mutt:it-v. so this statement either holds cm it does not Inthe language of statistics. we have given a confidence interval for thelillk110111/ deftlillillititie quantity ir,'.(Z2 ) As described, the procedureproduces a confidence interval whose upper limit is either the bound0 593. sm. that we aim for. or infinit y, if Ill < 21.4)

Narrow confidence intervals for mars or her critical probabilities havebeen obtained by .Rionlan and \Valters [2006]; Ica example, p;1(11) E[0 6965..0 6975] with 99 9999 1X confidence. where is the hexagonallattice

As usual in statistics. we must be a little careful in generating confi-

dence intervals: Ito example it is not legitimate to per fornt various runs

166 Ethmaling critical probabilities

of the sampling procedure with various parameters, and only use oneresult In practice, this is not a problem for two reasons: finitly we canperform as many • durnuty' Inns as we like to get an idea of parametersthat are very likely to work, and then one teal run with these parametersAlso, it is easy to get a failure probability of 10- 9 , say, for each run Itis then legitimate to perform 1000 runs with different parameters, andtake the best bounds obtained: as long as the probability that each rungives an incorrect bound is at most 10 -9 , the final bounds obtained stillgive a. 99.9999% confidence inter val for the critical probability.

There is another pitfall to bear in mind when implementing the prob-abilistic procedure described above: we have implicitly assumed thata source of random numbers is available.. In practice, for this kind ofsinndation one usualli uses a pseudo-random number generator Notall the widel y used ones are sufficiently good lb/ this pm pose Indeed,the current standard generator I t anclourfl used with the programminglanguage C is not

For example, using this generator we obtained estimates for 1' 10 (0 731)of 0 8661 ±0.000 2 and 0 8631+0 0002, depending on the order in whichthe random states of the sites were assigned (The turc:er tainties givenire t wo standar d deviations ) Using the much better I Merserme Twister'generator MT19337 WI ititen by Matsumoto and Nishimura, we obtainl i o(0 131) = 0/8630 ± 0 0002 Note that this (presumably) true value issmaller that 0/8639, /i hile one of t he values obtained using I andom() islarger. in generating t he confidence intervals given above, we used the

lersenne Twister. Of course, one can re-run the same procedure tr itlra different generator. or with I n rte' candour numbers obtained horn, forexample.. quantum noise in a diode

The methods described above can be easily adapted to give good up-per and lower bounds on p4!(.(1) and pII(A) for any of the Archimedeanlattices A. 1701 lower bounds. to prove that percolation does not occurat a particular value of p one can use 1-independence as in the proof ofexponential decay of the volume in Chapter 3.

For another approach to lower bounds, let S„(0) be the set of sitesat distance n 11010 the origin, and let N„ be the number of sites in5„ (0) that may be reached from 0 by open paths using only sites withindistance 11 of 0 Lemma 8 of Chapter 4 (or its equivalent, for bondpercolation) states that if, for some n, we have 1/1/,(N„) < 1, then thereis exponential decay of the radius of the open cluster Co, which certainlyimplies that p < pc . In am- lattice, for any p < Menshikov's 'Theoremimplies that: a p (N„) — 0 as n ^ ob , so arbitrarily good bounds may in

6.3 Oriented percolation on 22167

pr inciple be obtained in this way As before, exact calculations of Ep(Ar„)are impractical except for vet y small but estimates with rigorous errorbounds may be obtained for larger re This observation is applicable toany finite-type graph, including any lattice in any dimension

For bond percolation on a planar lattice A with inversion symmetry,to find a lower bound on plc' (A) = 14.11 (A) = pi (A), one may find an upperbound on pl„)(A*) and use Them ern 13 of Chapter 5, which states thatp l,(A)-t-p 1JJA*) = I. Similarly, for site percolation, we may find an upperbound on p4(AX) and apply Theorem 14 of Chapter 5 This approachseems to give better results in practice The reason is that it is easier toestimate the probability of an event by sampling than to estimate theexpectation of a random variable that, might in principle take quite largevalues occasionally

6.3 Oriented percolation on Z2

The study of oriented percolation, and, in panic/dar t bond percolationon the or iented graph 23-, is a major topic in its own right.; see Durrett[1984] for a stir vey of the early results in this area. iented percolationis in general much harder to work with than unoriented percolation,a fact that is reflected in the difficulty of obtaining good bounds on

Z 2 ) Indeed, Durrett described the question' of finding a sequence

of rigorous upper bounds on 1 .21!1 ( Z 2 ) that decrease to the true value as

an important open problem His bound, pi'( Z 2 ) < 0 84 , Was very fat

from the tower bound of 0.6298 obtained by Mar [1982]. In contrast,for lower bounds, it is easy to produce hounds that do tend up to thetrue value Indeed, let N„ be the number of sites on the line x y =

that may by reached from the origin. If Eih,(N„) < 1 for some n, then

Lemma 8 of Chapter 4 shows that p < pfi ( Z 2 ) < Indeed,

Hanumnsley [1957b] deduced that pl I (Z 2 ) > 0 5176 from the fact thatEb (No) = 4p2 — 11 1 . Of course. by Nlenshikov's Theorem,Z -) =-p!iFE 2 ) =1.4,'(72), say

There have been many Monte Car lo estimates of pN Z 2 ): such resultsare not our main focus, so we shall give only a few examples, rather thanattempt a complete list: Kertósz and Vicsek [1980] gave the estimate

pcb ( 2.,Y ) = 0 632 ± 0 004. Dinar and Bar ma [1981] reported p[ (G2)

0 6445 ± 0.0005, and Essam, Gultruann and De'Bell [1988] ( Z 2 ) =0 644701 ± 0.000001, for example

168 Eqinading et Theo! probabilities

Bali:der. Bollolgis and Stacey [1993: 199-1] gave Indict tippet bounds

ou p l:( Z 2 ) using the basic sir ategv of comparison with l-independent

percolation. Init ill a much mow complicated WaV than that described

iu the previous section. Their approach does give a sequence of rigorous

upper bounds tending down to the true value As Hie alguntent is lather

involved. NiAi shall give onl y an outline

\Alien consideting oriented percolation on OW aim is to decide

tot which p Ha' percolation probabilit y O(p) 00 (p) is men-zero In

doing so. we was of cows() restrict mu attention to that prat of 2)2

that Mak C011telVabh : LC l eached hoot the (nigh). namel y the positive(radian (2 The basic idea is to use independent bond percolation on

Q to define a le-Sealed I-independent bond percolation on Q To (I()

this, We choose parameters b and h. and at 'fume rhombi with II+ I sites

along the bottom-lett and top-tight shies as in Figure 3 Mow precisely

Pigmy: 3. Rhombi with sites along the base (bottom left)) and the top (tipperlight) II each t hondtns is replaced by all oriented edge and I he top andLot too t of each rhombus be a \et tux the tesulting or iented pupil is isomm

to f 2 In realize the rhombi ate lagged : each contains exact IN :I sites ham

each of t la: 9 la% epf of 772 that it meets

wlft iug = Ei : .r. y > ± q = for bagel for

each site r iu L 1 we choose a set 5,, of consccutke sites in L h(t+. 1) . so

that the sets Se ate disjoint Also. fin each bond 7 E Q we choose a

(toughl y rhombic) subgtaph of Q in such a wry that R R- and P7

ate disjoint whenever 7 and f me bonds of Q. that do not sha l e a site

\\ :e legnite that it e = UP then the bale of R T , defined in the nattu al

wr y( is exact 5„ , turd the top of 117 is 5,

3 Oriented percolation On V 169

As before . WI it e Cit for the open out-clunk] of the i e.,the set of sites that may be reached fron t the might open (oriented)

paths. It is easy to show that there is a pu = ( -) < 1 such that,

for any l-independent bond percolation measure F on - iu which eachbond is open with probability at least p H we have P(rof >the ingument is vets similto to the moot of Lemma in Chapter 3Indeed, if Co is finite, then there is a dual cycle S surrounding the origin..such that every oriented bond star Ling inside S and ending outside isclosed As shown in Chapter 3 there me, ve tt- ctuclely, at most :te321-2

dual cycles S of length 2( surrounding the or igin For each, there is aset E of exactl y f oriented bonds starting inside S and ending outsideTwo horizontal bonds in E cannot share a site, and nor can two verticalbonds in E. so there is a set C E consisting of fl/121vertex-disjoint

bonds The fill-probability that all bonds in ate closed is at most(1 — ) 1/2 so the expected number. of cy cles with the required propertyis at most

which is less than I if p i = 997. sa t Note that the expression aboveis exactl y the same as that appearing i l l the mool of Lemma ll ofChaplet 3, not b y coincidence By counting more car Mont the cycles .5'that ma t actually arise as the boundary of Co. as in Balister ,11n1 Stacey [19991, one can obtain a bet ter bound on pi

If we define an event C:( U T- ) depending on the states of the bonds in

fly so that an infinite or iented path of bonds 7 for which C(R7 ) holds

guar antees an infinite open oriented path in Q. and if. (or some p. wecan show that Pp (G(RT )) > pr for me t y(-T then it Follows easily that

TT,i ( -) < p Howevet , it is not easy to see how to define G(R7 ): piecingtogether paths in this context is much harder than in the unot iented case.

The actual atgm/rent of Balist et. Bollobfis and Stacey is notch moresubtle Choosing the regions fly so that they are isomorphic to oneanother, the y seek a Wont—trivial up-set A C 'P(11/± II) with the Followingproper ty: let i = 171 t, and consider the random set U of sites in S„ thatmay he leached nom the otigin b y open oriented paths. This set maybe naturally. identified with a subset of [b-f- I] Let V be the set or sitesin S, that may be reached horn U br open paths in again this setmay be identified with a subset of [b+ II The condition [conked is that.

1 70 Estimating el die& pmbabilitie5

for some p p E (0, l), whenevet U E A, then

17 1,(1 .7 E A U) > p (3)

There is a further restriction, that A be symmetric under the operationi b + 1– I. so that the identifications described above may be madeconsistently.

Bollobas and Stacey [1993] show that, under these assump-–''lions, one can construct a re-scaled oriented bond per colation on L-

in which each bond is open with probability at least p, and which iseffectively 1-independent (Mole precisely, given the states of the bondsbelow sonic layer, the bonds in that layer ate 1-indepertdent..) They showthat, if p > (3 – 07) )/2 and p > 1 – (1 – p) 2 , then such a process dom-inates the original independent process Using Menshikov's Theorem,one can then deduce that p> -).

In summar y, if one can find a suitable region R., au up-set A, and

a p > (3 – V17) )/2 such that (3) holds with p = 1 – (1 – p) 2 I'm all

U E A, then p is ' in tippet bound fin p l!( Z 2 ) Note that this does notcon espond to a direct construction of a 1-independent percolation modelas in the unotiented case Such a direcl construction could be achievedby defining 0(1?7 ) so that if G(Rn" ) holds and U E A then V E A;indeed, one could take this as the definition of 0(117 ) But then thecondition that 0(1 7-) ha t e probability at least p i amounts to

Pp(U E A V E- A) >

which is infinitelt stronger than (3) with pIn order to apply the method above, it seems that, we have to make

a large rininiter of choices: lust, we have to choose a suitable legion P.

and a probability p Then, we must also choose one of the 2E-4'.fP/t/n

possible up-sets to test Fmtunately, there is an ;lige/Ulm' given byBalistet, Bollob 'is and Stacey [19931 that, for a given I? and p, testswhether there is am] up-set A with the required property (and findsthe maximal one if there is) Using this algorithm, together with the

guluent s outlined above, they moved that p:( L 9-) < 0 6863. They alsoshowed that. as in the case of man iented percolation, the method givesarbitrarily good bounds with sufficient computational effort This is amuch harder result than the (almost trivial) equivalent for fluor ientedpercolation described in the previous section

Using a more sophisticated version of the argument, involving a morecomplicated 'eduction how the dependent percolation to independent

6 3 at iented percolation Oil Z2171

percolation. Balistei Bollobas and Stacey 1199-II obtained tare following

hounds.

Theorem 4. The critical piobabilitics for oriented bond and site perco-

lation on the square lattice satisfy

pc1;'(27: 2 ) < 0 6735 and pj"(1%;( 72 ) <0 7191 q

As in the case of umaiented percolation, a (simpler) version of the o-

p/molt; just described may be adapted to give bounds with 99.99%) con-

fidence!, and fa/ stronger bounds can be obtained in this way with much

less computational effort Indeed, Bollobtis and Stacey [19971 showed1 , -7

that pc ( < 0 6-17 with 99.999967% confidence. This bound is very

close to the value of appioximately 0.64415 suggested by simulations;

Liggett (19951 used a beautiful and totally different approach to bound

p,/,'( Z 2 ) and pr.(2-), based on au idea of David Williams, who conjec-

t ined that p(1 '( Z 2 ) < 2/3. To describe this, note that ()dented peicolit-7-/,

lion on corresponds to a Mar kov chain in a natural way Recall that.

the ith layer is the set L i = {(:r, y) E 1Z2 ) : t y > 0, a' -F y = t.} The

set TI t of sites in L i that be reached from the origin by open paths

depends only on and on the states of the bonds between and

L t Obi oriented bond percolation). 01 the states of the sites in L i (for

site percolation) More explicitly. let us code R t by t he r-coordinates of

the points it contains, setting

A i = : (r t t — :r) may be reached from 0 by an open path}

Then Ao = Fat bond percolation, given ./14 , the probability that :I! E

A,+1 is 0, = = 2p—p2 according to whether lA t ofx-1, j is 0,

1 01 2. Furthermore, given A t , the events E A t+ , } are independent

for different c;. For site percolation, the jiMarkov chain is the same,

except that p i = p2 = p. Thus, it is natural to extend the definition of

the Maticov chain (A t ) to any pan/meters 0 < < 1 and 0 < pa < 1

Note that this Markov chain has stationary transition probabilities: the

distribution of 1 t t , given that = A is the same for any t. Of course,

this Nlarkov chain underlies any analysis of ot hinted percolation; lot the

approach described above, we did not need to define it explicitly Liggett

j1995] !moved the following result.

Theorem 5. If the parameters mid ha satisfy the inequalities

1/2 < < 1 and (1 — ) < p2 <1, (1)

17 9 Estimating critical probabilities

Then the chain (A 1 ) defined above satiqics Pec't : A l = (0) > 0

Em bond pcs ("dation. 'slagpm p and Ps = 2p — ti conditionsof I ()mew 5 MU satisfied fin p > 2/3 For site percolation. whewpu = Ps p, the relations (-I) hold fin art's p > 3/1 Thus Liggett'sresult in t mediatels implies the following bounds

Theorem 6. The (atheist probabilities lot minuted bond and site perco-lation on the square lattice satisfy

< 2/3 and p.( 2 ) G

Given a finite set A C E. let denote the landont set obtained byminting the M iukov chain lot k steps stinting with the set A Thus, ifA = {01, then .4" has the clistlibut ion (if A„ The basic idea of Ligget t'spool is as follows: we hope to define a Function on the finite subsetsA of Z so that II(0) = I awl M I all A 0 lye have fl(A) < 1 and

EgH( A I )) < H(A) (5)

Of course whether such int I7 exists depends on the parameters p i andpo of the :\ farkov chain If such an 11 does exist, then from the Nla i km"

opei tv awl (5) we have

E(B(.<I„)) =E(la(lLi„ A„_ 1 ))) E(H(..1„_1))

< (AO) = 11({0))< I

lira civets i s This implies that the pelcolat ion incites); is supple' it ical:otherwise. we would Inge = 0) — 1 as n — x As 1-1(0) = I awlfl is bounded. this would imph /1(A„)) — I. a contradiction

Liggett [1995} showed that. it the conditions (-I) hold then an H withthe I equited mopet ties can be found The weight fitnet ion 11 used byLiggett is Lather complicated

To describe Liggett's argument. let T be a minion 's variable takingpositive integer values. with E(T) < and let I( be the stationary' encash' meastue on sequences q E 10 1 t- associated to h Thus. thediSt / Haitian of I he landain secnwnce rt is itr yltiant under tt mishit ion. andthe gaps between successive Is a t e independent and have the distributionof IT Lhe weight 11(A) is defined as the ic-probabilit y that l i (y) = 0 formet v .1‘ E) A. Note that this etamint y satisfies 11(0) = 1 awl 1-1(A) < tFm = 0. Of tom w het het the kev equation (5) is satisfied dependson the choice of

(1 (bleated pelfolation on Z 173

\\j i lting F(n) for ENT > a). Liggett shows that (5) is satisfied in the

special case that A is an inter val if and only if

p 2i 110 + I) = — .17(k)F(n — k) a1)2F(7I) (6)

fin ever > 1 El he derivation of this condition is relativel y simple:

roost of the work lies ahead. First. one must show that. if (-I) holds.then the solution to (6) with F(1) = t makes sense. i e . cooesponds

to F P(T > n) fin some T I iris amounts to showing that 11(a)is decreasing and that .Y[ ' n F(n ) L(T) < x Second. OM must show

that (5) holds not only for inle t als. but also for atHutto sets .4 Even

though the Mat kov chain operates on disjoint Mor t vals sepruatelv.

some sense, due to the complex definition of If t his is by no means a

simple consequence of t he interval case Indeed. t he tuguments of iggett

[1995] for both steps are far horn east

Let us rcuuulc that oriented percolation on 2 fray be thought of asa discietc-t hue version of the contact process on E Indeed. Liggett sproof described above is similar in outline to an argument of Holies and

Liggett [1978] ghing an upper hound for the critical parameter of thiscontact process As noted 1)5 Liggett [1995] it is much harder to tututhe outline into a proof for oriented percolation than tot the contactprocess. due to the discrete nature of the process.

At tire tint(' of WI iting. Liggett's Intim ( ' < 2/3 is the bestrig-

orous upper hound on p l/ (L 2 ) Liggett notes. however. that Iris method

cannot he pushed further: the solution C (a) to (6) above has the re-

quired properties if and only rf t he conditions (i) are satisfied. In con-Oast. the method of Batiste/. Bollobtis and Stacey [1993: 199-0 gives

arbitraril y good bounds with increasing computational effort busat some point it will become feasible to obtain a better bound in thisway . Indeed. this nay have aheadv happened: the amount of comput-ing power available now is much larger than it was when the boundp[[( Z1 2 ) < 0.6735 Willi obtained in 1994 FM site percolation. the boundof Batiste/. Bollobris and Stace y [1994] given ill Ille01 CM alreadybetter than 3/4

As Liggett le/narks. Lincoln Chaves pointed out to hint that Theo-

tem 5 gives bounds on the critical probabilities km percolation on ori-

ented lattices ot her than 72 twitted. let TI he the oriented hexagonal

lattice shown in Figure 4 Taking ever v second column of sites as a laver

L I as in the figure. a given site in tr. j. j runs be reached only hoo t one

Estimating critical probabilities

t :3

igme 1 'The portion of the oriented hexagonal lattice that may be flinchedloom the origin Considering the sets of points in evert . second column t liatmay be reached from the origin by open paths leads to a simple Marlcov chain

of two consecutive sites in 1, 1 Further mote, all routes from L i to a site

E L t+1 are disjoint from all routes from L I to any other site to E

Hence, both for site and for bond percolation, given the set PI of sites

in .L 1 that may be reached horn the might, each site E Lg +1 is in R1_1

independently of the other sites. Thus, the sets Rr may be described by

the same Markov chain used in the case of Z 2 with

P i p and P2 = P(2P P2)

for bond percolation on H. and

Pl = P2 =

for site percolation ou H Consequently. Theorem b implies the bounds

P I: ( 11 )3t3

mid /PM 9

6 4 Non-z iymotts bounds 175

64 Non-rigorous bounds

It seems that the critical probabilities for even very simple graphs canbe determined exactly only in a small number of exceptional cases Inthe light of this, it is not surprising that a huge amount of work hasgone in to estimating such critical probabilities. Here we shall describebriefly some of the techniques used, and mention a few of the older anda few of the more recent papers on this topic: a complete survey of thisfield is beyond the scope of this book

??lost non-rigorous estimates of critical probabilities for lattices atebased on Monte Car lo techniques, i e., on simulating the behaviour ofpercolation on some finite lattice However, there are many differentways of extrapolating front such finite simulations to predict the criticalprobabilities

For simplicity, we shall consider site percolation on the square latticeZ2 Let A„ be the finite portion of this lattice consisting of a squarewith n sites on each side If A„ is any event depending on the statesof the sites in A.n , then we May estimate Pp (A„) = PE;24,(A„) in theobvious way: for each of N trials generate the states of the sites in A„at random, according to the measure P. and count the number M oftrials in which A„ holds

The problem is this: how do we estimate p„ (Z2) = f(Z2 ) fromthe probabilities of events depending only on the states of finitely manysites? One possible approach is the following Let H,, = H(A„) be theevent that A„ has an open horizontal crossing From Menshikov's The-orem and Lemma 9 of Chapter 5, we know that lim„_, P p (H„) = 0 ifp < Pt , while lim„_.„ „(H „) = 1 if p > p„ For each ti re functionPp (Hn ) is increasing in p. Hence, as a gets larger and !anger, this flute-Hon increases from close to 0 to close to 1 in a smaller and smaller 'win-dow' ar ind pc In particular, the value p(n) at which Pp(„)(11„). 1/2,say, tends to pc. By estimating Pp (II„) for various values of p, we canestimate p(a). By choosing n as large as is practical, or by consideringseveral different values of n and extrapolating, one can then estimatepc = lim„_,p(n). A problem with this approach is that, even for asingle value of a, we must estimate P„(11„) for many values of p

A computationally efficient version of this method was developed byNewman and Ziff [2000; 2001], based on the following simple idea.. LetQ„ be some random variable that depends on the states of the sites inA„, whose expectation we wish to estimate For example, C2„ could bethe indicator function of the event LI, so EgQ„) p(rin). If there

176 Estimating critical pan/abilities

ate N = IV(P) sites in A„, then. writing X for the random number ofopens i tes in A. we Ittoe

E i,(Q„) = E(Q„ = ittliPp (X = k) Eb( Ar.p. fit)E(Q„ X = kk=t)r-4)

where lt(N,p, k) = (fiT)pk (1 - p) N-k is the probability that a binomiallandorn variable with parameters N and p takes the value k -111/IS,

to obtain (non-independent) estimates of Iii i,(Q„) for all values of p. •

suffices to estimate E(Q„ X = k) for each value of k. After conditioningon = k. the set of open sites is a iandon, subset 01; consisting of k sitesof A. distributed unifor nily oyez all ( tin such sets \Ve call efficientlygenerate a sequence (0,; )ii of random subsets with the light (marginal,i e . individual) distributions by considering a random set process: startwith. = 9, and generate (9 t .i. t horn 0, by adding a site of A„ \ 0,

chosen at random, with each of the N - l sites (Totally likeli II wegenerate It such sequences independently, for each k we may estimateE(Q„ X = A') as the me/age of Q„ over the I? samples for Or,- wehave generated Combining these estimates as above gives estimates ofEi,(2„ ) for all p simultaneously

Often, the behaviour of Q„ varies in a simple way as the state ofone site is changed how closed to open. in which case this poet:Mire isen fast For example. taking Q„ to be the indicant' function of the

event H„ = 11(11„), the dues of Q„ for an entire sequence (Or,) Call hecalculated in lincep time (in the number of sites), using a 'union/lied'alp/Hun to keep track of the set of open clusters at each stage Thisenables Der y accurate estimates of the curve ili i ,(99„) to be made even lot(mite huge n, which in turn means that ',Mean be estimated accuratelyUsing a variant of this method, considering the event than some opencluster on a toms 'wraps around' the toms in a certain way, Newman

and Ziff report the following estimate fon pc = gt(Z2):

li„(E") = (1 5027-16'2110 00000013

Flowevm. there ate two problems with the approach just described Oneis that the estimates of E i,((2„) kit different p ate not independent 'Thismakes the statistical analysis somewhat Made/. A ouch mote set ionsmoblent is the effect of Jinn C-siZC scaling': we know that p(p), sa i tendsto pc , hut not at what rate. It is believed that

17)

6. 1 Aton - 1.1gormt, bow, tb 177

lb/ 501/K/ ("011S/ Hifi o: this would follow hoot li re existence of a certaincritical exponent': see Chapter 7 `the existence of this exponent isknown only for site percolation on the triangular lattice, where pr-in any case known exactl y. Bleu if (7) were known, it would not, helpto give rigorous hounds: such a relation still tells us nothing aboutthe relationship between p„ and the finite set of Values p(n) we haveestimated, as the o(I) term could he very huge (or small) for smallvalues of n.

In 8111111/011 V, Monte Carlo 'nethods as described above produce esti-mates for pt. that do (70 /1/1/1 ge in pi obabilitt, to the tun e value, as the size

the finite lattice studied and the number of simulation 111/IS increases-loweyer while the statistical en of can he analyzed. the r ate of conver-

gence to pc is unknown Thus an y HIDt bounds given by these methodsamount to an educated guess as to the final uncertainty This contrastswith the rigonms 99 99% confidence intervals obtained by consider ing1-independent: percolation, where the only stance of (trot is statistical,

tund the error probability lot a given bound can be determined exact ly

To show that this problem of estimating the uncertainty is a realone, note that there is 80111() disagreement about the rate of convergencefor the Newman -Ziff method described above. Newman and Ziff [20(11]say that the finite-size error decreases as N- 11,18 , when) N is the num-ber of sites in the finite lattice studied however, the numerical Je-suits of P/I/ [2005! stiongl ) suggest that the r ate is closer to_y - fir There are also many examples of reported :results contradictedby later estimates, or even by rigorous results An example is the es-innate 6'1 ( 7,, 2 ) = 0 632 ± 0 00+I given In keitesz and Vicsek [19801 wementioned cattier Nonetheless, these methods do give very atell/ ateestimates oh pc : the trouble is that one cannot be sure how accurate!

7

Conformal invariance - Sn mov's Theorem

The celebrated 'conformal invariance' conjecture of Aizemnan and Lang-lands, Pouliot and Saint-Aubin (199,1] states, roughly, that if A is a pla-nar lattice with suitable symmetry, and we consider percolation on Awith probability p MA), then as the lattice spacing tends to zerocertain limiting probabilities are invariant under conformal maps of theplane P2 C This conjecture has been proved for only one standardpercolation model, nameh independent site percolation on the triangu-lar lattice The aim of this chapter is to present this remarkable result ofSmitum [2001a; 20014 and to discuss briefly some of its consequences

In the next section we describe the conformal invariance conjecture,in terms of the limiting behaviour of crossing pr obabilities, and presentCardy's explicit prediction for these conformally invariant limits inSection 2, we present Smimov's Theorem and its proof; as we give fulldetails of the proof, this section is rather lengthy. Finally, we shall verybriefly describe some consequences of Str u t nov's Theorem concerning theexistence of certain 'critical exponents '

7.1 Crossing probabilities and conformal invariance

Throughout this chapter we identify the plane P 2 with the set C ofcomplex numbers in the usual way A domain. D C C is a non-emptyconnected open subset, of C. If D and D' are domains, therm a conformal

crap from D to D' is a hijection : D D' which is analytic on D.

i e., analy tic at every point of D Note that • l is then analytic on D'.Locally, a conformal map preserves angles: the images of two crossingline segments are curves crossing at the same angle; this is why the ter mconformal is used By the Riemann Mapping Theorem( (see, tor example,

7 1 Crossing probabilities and conformal invariance 179

Duren (1983, p 111 or Bear don (1970, p 2061), if D, D' C me simplyconnected domains, then there is a conformal Map from D to D'

Roughly speaking, co/Mutual invariance of critical percolation meansthat certain (random) limiting objects can be defined whose distributionis unchanged by conformal maps. Here we shall only consider a moledown-to-earth statement concerning crossing probabilities For this, weshall need to consider domains whose boundaries are reasonably wellbehaved

We write D for the closure of a domain D Let us say that a simplyconnected domain D is a Jordan domain if the boundary D \ D of D isa Jordan curve, i e the image I of a continuous injection 2 : T C,where 7 = PlZ is the circle, which we m ay view either as [0,1] with

and 1 identified, or as {z : = We shall write F(D) for theboundary of .D, and 7 = 7(D) for a function 7 as above, noting that 7 isunique up to parametrization.. By a k-ma,11,:cd domain (D;we mean a Jordan domain D together with k points Pi , P), , Pk onthe boundary of D We always assume that as the boundary F(D) is tra-versed anticlockwise, the points 121 appear in this order We shall oftensuppress the mar king in our notaticm, writing Dr, (Of (.19; Pk)

Given a marked domain Dr; , we write A 1 = (D,,.) for the boundm y

arc from P1 to Pi+i (in the anticlockwise direction), where we includeboth endpoints, and the indices are taken modulo k; see Figure 1

Fiume 1 A 4-nnuked domain = (D; Pi , Pi, Ps, PI)

Let A be a planar lattice Given a real number S > 0, by SA wemean the lattice obtained by scaling A by 6 about the origin Thus, forexample, OZ 2 is the graph with vertex set {(So, : b E Z} in whichtwo vertices at distance rS are joined by an edge Suppose that we havean assignment of states, open or closed, to the bonds or sites of SA If

= (D; Po, P3, PI) is a 4-mar iced domain, then by an open crossingof D from. A i to A3 in SA, we mean an open path non r., t. in OA

JI

11111-11

I

1 11: I 1' i I [ I I r I

--m- - , , ,

, , R - ,

,,, , , ,, , 11 I I I I

I 1: Ill

11 II 11111

1 [ 111111'

II!

180 Conpain& invaDaaer Smanov s Theorem

such that /7 1 , cr_i lie inside D 1 ,0 and id ale outside D. and t he line

segments eon and et _ i n meet the arcs = (Dt land A 3 = A30).0,

respectively When the context is cheat, we IllitY omit the references to

the arcs .1 1 and A3 and to the lattice SA

The quintessential example of a -1-mat Iced domain is a rectangle with

the comers marked Let D = (a,b) x (e..d.), and let be the vertices

of the rectangle. D, labelled as in Figure 2 If A is the square lattice

P1 P2

Ir

Figure 2 A path P in the lattice 0::1

D I The path P is an open (dossingopen ciossing of D. 1 in site (a bondInn iZOIdal n ossing of t he act angular

crossing a reel angular •I-marked domainit all sites or bonds ()I P are open Anpercolation on 6:2 is exactly an opensulnpaph l? of <5222 shown in the figure

Z1 , then the subgraph I? of SA induced b y the sites in D is a rectangle

in ST2 in the sense of Chapter 3. For n < — . d — c}. an open

clossing of = (13; Pl . Pi 8.. P 1 ) in the site or bond percolation on

SA is exactly an open horizontal crossing of the rectangle II' obtained

front by adding an extra column of points to each vertical side \Ve

shall see later that the precise manner in winch we heat the boundary

is not Minot tant: we could have defined an open crossing of at as an

open path inside .0 starting at a site 'adjacent to' A I awl ending at a

site 'adjacent to' A3. for example

Let be a 4-marked domain, iind A a pl anar lattice Considering

either sit e on bond petcolation on A. for b > (1 mid U < p < 1, let

f--;) (D ) . A. p) 11 1 '1 ,(D. ) has an open (dossing in SA),

ehere IF,, is the site or bond percolation measure on SA in which sites or

bonds a l e open independently With probability p 01 course. as usual we

should indicate in the notation /d ) (D.I . A. p) whether we consider site in

howl per (dilation, but this will not be necessin : shot tly. we shall testi ict

7 1 emssing probabilities and conformal invariance 181

our attention to one specific model. site percolation on the iangularlattice

p < Th . = p 1 0), pr (A) is fixed. then it follous from Menshikoy's

Theorem (see Theorems 6 and 7 of Chapter -I) that Ps(D 4 , A. p) —tt 0 as--, 0 Indeed. the arcs Ar = Au (D 4 ) and fla .51:1 (D4 ) ate separated

1w a distance // > 0 Roughly speaking, as a --, 0, if D i has an open

G lossing then one of the 0(1/6 2 ) sites near A I must he joined b y anopen path to a site at graph distance (1)(/./O) = e(1/6). By Mensbikov'sTheorem and exponential decay below p t (Theorem 9 of Chapter -1) thepmbability of this event is 0(1/6 2 ) exp(-6(1/6)) = of I) as /5 0 (Thehound 0(1/52 ) on the number of sites irm D 'adjacent: to' A t is of coutserather crude: note, howevei, that this munbei need not be 0(1/5),A l has a fractal structure fin example ) Similarly. considering the dual

lattice (for hood percolation) or matching lattice (fin site percolation).

it follows that Ps(DA . A. p) 1 as 6 0 with p > pc. fixed This begsthe question of what happens when p =

Flom now on, we take p to lie the critical probability = ThjA), andmite /),;(D. 1 , A) for .Ps(Dj . A. pc ) To be pedantic, we should indicatewhether we consider site or bond percolation. but we shall not do so.\1'c have seen in Chapte i 3 that it D. 1 = (a.b) x (c, (/) is a rectangle withthe corners marked, then the Russo-Seymour-Welsh Theorem impliesthat there is a constant: o(D.1 ) > 0 such that

<Po(D.I,'") <1— o (I)

holds fior all sufficientl y small 6. (hi fact. a < inin(b — d — c) will do,since this condition on a enmities that D must contain points of OZ 2 ) Acorresponding statement for any lattice with suitable symmetry can heproved along the same lines Using Elm ris ls Lemma, it. is not hard todeduce that (1) holds lot any A-marked domain D.i.

In the light of (1). it is highly plausible that for any lattice A andany 4-marked domain D. 1 . the limit lintszo P,t (D. 1 , A) exists and lies in(0,1). Langlands. Picket. Pouliot and Saint-Auhin 119921 studied time

behaviour of this limit assuming it exists, by perrot/Mug 1/1/11/CIical ex-pet intents with rectangular domains Dr, for site and bond percolation

on the square. triangular and hexagonal lattices These experiments

suggested to them that the limiting dossing probabilities .P(D. I . A.) areuniversal, i e. independent of the lattice A (This is au oversimplifi-

cation: in general. one must first apply a finest transformation to the

lattice A: for the cases listed. this is not necessary) Aizenman then

suggested that these crossing p i obabilities should he confin wally inyar

182 Canfonital thou] lance - Strtirnoo's Theorem

ant; supported by additional experimental data, this was stated as a'hypothesis' by Langlands, Pouliot and Saint-Aubin [1094]

Conjecture 1. Let A be a. 'suitable' lattice in. the plane, and let D., =_

(D; p3, ) 4-marked domain Then the limit.

P(D. 1 , A) = lim Po(D.,, A)

exists, lies in (0,1), and is independent of the lattice A Ful thermal

and ry, nit cmrformally equivalent 4-marked domains, then

P(D.1 ) P(D'4)

Let us spell out the meaning of conformal equivalence in this con-text. Car atheodor y's Them ern (see, for example, Duren [1983, p.12], orBeardon [1979, p. 226] for a proof) states that if .D and D' are Jordandomains, then any conformal map f from D to D' may be extended to

a continuous map f from D to D' As f <I may also be so extended, it

follows that f maps the boundary of D bijectively onto that of D' Let

/34 = (D: (P)) and D', = (D'; (PI)) be 4-marked domains. Then al

and D!, are conithmally equivalent if there is a conformal map f from

D to D' whose continuous extension to D maps Pi to PI for 1 < i < 1

Note that the crucial condition is that P, is mapped into PR we always

have conformal maps ft urn D to D'To describe the scope of Conjecture 1, let us define what we mean by

a lattice: a two-dimensional lattice A is a connected, locally finite graph

A, with V(A) a discrete subset of R 2 , such that there me translations T,

and 7",, of R2 through two independent vectors m and va each of whichacts on A as a graph isomorphism. Note that we do not require A to be aplanar graph. Also, as in the case of planar lattices as defined in previouschapters, the vertex set 1 7 (A) need not lie a hit tice in the algebraic sense:in general, V(A) is a finite union of translates of an algebraic lattice hrthe conjecture above, a 'suitable' lattice includes any two-dimensionallattice with rotational symmetr y of order at least 3 This includes thesquare, triangular and hexagonal lattices, for example

One cannot expect conformal invariance for site percolation. say, onan arbitrary lattice A. Suppose, for example. that A is obtained fromZ2 by applying the shear described by a matrix AI. Then conformalinvariance for Z2 would imply that P(D 1 , A) = P(Do, A) whenever the

domains M- 1 1), and Do are conkn malty equivalent This is not

the same as D I and Ds being confer/natty equivalent. The conjecture

of Aizetunan and Langlands. Pouliot and Saint-Aubin [1994] states that

7 I Crossing probabilities Wad conformal inuariance 183

for any two-dimensional lattice A, there is a non-singular matrix AIsuch that P(D, AI A) is confonnally invariant, and equal to P(I),Z2),say They also state that the same result should hold for many non-lattice percolation models, citing experimental results of Maennel forGilbert's disc model (defined in Chapter 8) and Yonezawa, Sahamoto,Aoki, Nose and Hori [1988] for per colation on an aperiodic Penrose tilingConjecture 1 is also believed to hold for random Volonoi percolation inthe plane (also defined in the next chapter); see Aizenman [1998] andHeMatnini and Schramm (1998]

As there are so many conformal maps, Conjecture 1 is extremelystrong. Indeed, a ny simply connected domain D C is conformallyequivalent to the open unit disc B, (0) Thus, any 4-nnu Iced domain isconformally equivalent to the unit disc with some four points zo, 23,21 specified on its boundar y The conformal maps from B 1 (0) onto it-self are the MObius transformations. Given two triples of distinct pointson the boundary of B 1 (0) in the same cyclic order, say (2 1 ,20,23 ) and

4), there is a unique ?bolus transformation mapping z i to 2;Thus, any 4-marked domain D., is equivalent to the unit disc with thefour marked points 1, VII, —1 and 2, say, where kir:: 1 and Im(z) < 0Hence, the equivalence classes of 4-marked domains under conformalequivalence may be parametrized by a. single 'degree of freedom'

Although, throughout this chapter, we work in the complex planeC LPf2 , we shall not often need to write complex numbers explicitlyThus, we reserve the letter i for an integer (as in z i above), and writej--7. rather than i

A natural parametrization of 4-marked domains is in terms of thecross-ratio: given four distinct points 2 1 , 23 , 2. 1 appearing in this cyclicorder around the boundary of Br (0), their cross-ratio )7 is the real num-ber

Z:3)(z2 Z1) = , E(0.1) (2)124 — 22)(23 — ZI

Mains transformations preserve cross-ratios; in fact, two markings ofthe domain B 1 (0) are conformally equivalent if and only if they have thesame cross-ratio Given a 4-marked domain ,D4 = (D; P2, P, PI),we may define the cross-ratio n(D. 1 ) as the cross-ratio of any markingof Br (0) conformally equivalent to D4: this is unambiguous, as any twosuch Matkings are themselves conformal! ), equivalent, so they are relatedby a Maius transformation and have the same cross-ratio With thisdefinition, two 4-marked domains are conlonnally equivalent

184 Conformal alma altar Sail! non s Theorem

if and only if 11 (1) 4 ) = Tints the conjecture of Aizennum andLanglands, Pouliot and Saint-Anbin [1994] states that P(D 1 .A) is given

by some function p(q) of the cross-ratio a(D,1)

Inspired by Aizenmatfs prediction of conformal invariance. Cindy[1992] proposed an exact contormally invariant formula 7 (D4 ) fin the

limiting crossing probability P(D I , A). i e . a formula "T (q) for the func-tion p(q). Using methods of conformal field theory, which, as he stated,are not rigorously founded in this context, he obtained the formula

3F(2/3)p(q) 7r(q) 2F, (1/3.2/3; 4/3; a)

r0/3)-

Het 2F 1 denotes the standard lopengeomett ic function, defined by

a(72)1,(H) z"

c(")

whe 3 , ( " ) is the rising Pictorial x i " ) = ,r(x +I )(x + 2) • • (x+ — I) (seeAbramowitz and Stegun [1966, p 556]) It is not clear whether Candy'sderivation can be made rigorous In fact, there is (essentially) only onecase where conframal iuvariance has been proved rigorousl y, namely site

percolation on the triangular lattice As we shall see in the next section.Smir nov's proof gives conformal invariance and Cand y 's formula at the

same timeThe case where Di is a rectangle with the corners marked is of special

interest Let D.,(/ ) be the domain ' (0, ) x (0,1), with the corners marked

as in Figure 2 Then the aspect ratio of .D, 1 (; ), i e the ratio of the

width of the rectangle Di (r) to its height, is I: the cross-ratio ti(D.1(t))

is given by some function q(r ), which is easily seen to be a decreasing

function flour (0, co) to (0,1) In par, ticular, there is one rectangle D4(1)

for every cross-ratio q E (0,1), so any 4-runuulted domain is confor wally

equivalent to sonic rectangle DI D ) Fin this reason', equivalence classes

of 4-marked domains w i den conformal invariance rue often known as

conformal rec taargies

As every rectangle has an axis of symmelny, mid any 4-marked do-main is cordon math, equivalent" to a rectangle.. Conjecture 1 implies theiucru ianee of P(D. 1 ) under reflections of the domain Di ; see Figure 3.

the cross-ratio ti (D. 1 ), and hence Candy's formula it(D 4 ), are

preserved by reflections, and hence by Mutiliolornor pink... ? maps, i e

jections "which preserve the magnitude but not the sign of angles

Given a il-manked .Jordan domain Di = (D: PI , P4). set =

(D; P. p m , .p„, Pi ). An open (" tossing of D. i e an open crossing of al

a Fi(o,

/ CIO ng plababilitias Ind con . .minal Huta/lance 185

---------

Figure 3 If f is a contramat map, then so is the map y : f Hance.a Turarked domain D m i d its minor image may be confornially mapped tothe same rectangle

[t om its first tO its third is simply an open crossing of D4 1101/1 A,to A., In the light of results such as Lemma 1 of Chapter 3, it is naturalto expect that P(1), 1 ) P(D,1) = 1 As q(D ,i) = 1 — q(D.O. one cancheck that Cm dy's for mula does predict this, i e., that

7(1 — q) = 1 — 7r(11)

By constructing an explicit: confirm/al map from the circle to A(l),one can use Cindy's formula to evaluate the value 7(D. 1 (1)) predictedfor P(D, I (t)) In fact, Cindy [1992] worked with the upper half planeU = {z : > (l} instead of the open unit disc. We shall not considerunbounded domains hen± but the definitions and results extend easilyto such domains under suitable conditions Ma rking four points 2 1 , z±

011 the !cal axis to obtain a T. /narked domain U, = (CT; (z i )), thecross-ratio t(l.i ) is also given by (2) For 0 < k < 1. let U l (k) he thedomain U with the foul points —14- 1 , —1,1, marked Then (1.1(k)11/10' be mapped by a Schwartz Christoffel transformation

(If

\/(1 —1-)(1 — k-/-)

to the interior of the rectangle I? w it II corners iihk(k 2 ) A:(k2)±k2 ) \,/T. whine

di A 00=

I V( — U)(1 —

186 Cortformal invariance - Spirt-t i tre's Theorem

is the complete elliptic integral of the first kind (Our notation herefollows Abramowitz and Stegun [1966. p 590) The same notationis sometimes used for K(u2 ) ) The aspect ratio of r is r 1(1,7)

2K(1c2 )//C(1 — k 2 ), so, to find 7r(D.di )), one can invert (minim jenny):this formula to find k; then 7r(D. I (r)) = 7r((.1.1(/,:)) 7r(q), where ri=tkU.0)) = (1 — k) 2 /(1 + k) 2 (In particular, this shows that tkr)

lAD4 (1)) is monotone decreasing in r )Starting from Cardv's formula, Ziff [1995a; 1995b] gave a relatively

simple formula not for r(D.1 (1)) itself, but for its derivative with respect

to :

20ITI (2/3)(fi k(D, 1 (0) — 1(1/3)2

(n±lit

The calculations outlined above illustrate an unfortunate propert)of conformal invariance: given two families of 4-marked domains, each:parametrized by a real parameter, even if both families ate 'nice' theconformal transformation from one to the other may II ansform the N.rameter in a rather ugh. way. Thus, one can not expect Candy 's for mukto take a simple for in for a given Mice' family of domains.

There is, however, one family for which Cm dy's formula can he writtenvery simply, an observation made by Lennart Carleson in connectionwith joint work with Peter Jones Let .D be the equilateral triangle in

IR2C with vertices Pi = (1,0), P, (1/2, 4/2) and .p, (0. 0), nnc

let Pi = (x, 0), 0 < <1, as in Figure 4

Figure 4 Cat leson s '1-marl

do

Car leson showed that for the special 4-nueked domain T;',, = (1);Cindy's formula takes the extremely simple form

= .r

(3)

7.2 SmintorC5 Theorem 187

As we shall see in the next section, it is in this form that Smir 1/0V provedCal(' \r'S ft/1/ .11111a for site percolation on the triangular lattice. Note that

relation 7r(q) 7(1 — rt) = 1 is easy to verify from (3) Indeed,permuting the labels of the rum ked points to obtain the dual domain

Tx* as before, Tjj is the mirror image of Writing ri n(21„) for thecross-ratio of L. , we have ti(13.*0 = 1— 0(1). 1 ) for any 4-marked domain,so

r(1 — = r(Trj) = 7T(7) -3.) — — X = — 7r(77,) = 1 — rt(q) (4)

The hypothesis stated by Langlands, Pouliot cord Saint-Aubin [1991]is a little more general than Conjecture 1 Their conjecture extends toevents such as 'there is an open crossing horn „4, to .,4 3 and an opencrossing from .40 to 24,/, and corresponding events involving any finitenumber of crossings of a domain whose boundary is split into a finite

number of arcs Conjecture I also has consequences that are at firstsight unrelated, namely the existence of various critical exponents Wesh a ll return to this briefly in Section 3.

7,2 Smirnov's Theorem

For the rest of this chapter we restrict our attention to site percolation onthe triangular lattice T. and the re-scaled lattices ST Throughout thissection we consider only critical per colation, i.e we study the probabil-ity measure F in which each site is open independently with probabilityp = p7.1 (T) = 1/2. As noted earlier . , Smirnov [2001a] (see also [20016])proved the remarkable result that the conformal irrvarlance suggested byAizenman does indeed hold in this case.

Once its walls have been breached, one might lave that the castlewould rapidly fall, i e that a proof for general lattices, or at least forother 'trice' lattices such as V, would follow quickly Although this waswidely expected, no proofs of other such results have emerged, except foran adaptation of Sruirnov's argument to a somewhat unnatural modelbased on bond percolation on the triangular lattice given by Chaves andLei [2006] It seems that Sruirnov's proof depends essentially on specialproperties of T

In this section we shall present Sinn nov's proof in detail. Let us notethat even the expanded version of this proof in Srnirnov [200114] is onlyan outline For the heart of the proof, showing that ceitaM functionsrelated to crossing probabilities are Inn-monk, it is very easy to dot theCS and cross the t's However, one must also deal with certain boundary

186 Conformal invarianceinvariance Smirnov Theorem

conditions Steil 110V 12001bi gives a suggestion as to how this should bedone, but it does not seem to be at all easy to tour this suggestion intoa proof. Bellina (20051 and independently Tad/ suggested consideringcertain symmetric combinations of Sinithov's harmonic functions; Befaro showed that tans greatly simplifies the boundary conditions, elim-inating the need to consider partial derivatives at the boundary(2005) has used these ideas to give an expanded account of Smirmov'sproof: nevertheless, even this presentation is far from giving all the de-

tails\\e shall move Stnimov's Theorem in the precise form below: the

original statement is somewhat note general in toms of the domains'considered The proof we shall present here is lather lengthy We followthe strategy of Stith nov. as modified by Beffar a Along the way, however,we prove the many 'obvious' statements that are requited

Theorem 2. Let D C C be a simply connected, bounded domain

whose boundary is a Milian curve Let Pi . 1 < i C -I. be distill()

boundary points of D, appealing in this cyclic order as I is travelsal,

so D I = (D: P. Pe. P3 PO is a -1-lnat hell domain. Let Ai(DA)

he the arc of F from Pi to PH. I . inhere p-, is taken to be PiThen

P(.0 1 ) = P,;(D,I,T) exists and is given by Candy's Mtmala where

p,s(D.,. I) is the probability that, there is an open crossing of D front

to A R in the ci Hirai site percolation on the lattice ST

The basic idea of the moot is to define cm taro functions that genet-

alize crossing mobabilit MS We regard as a 3-marked domain D 3 by

temporarily lotgettingthe fourth nuanced point P I , replacing the bound-

in V arcs :la. A4 1):1 a single arc A3 = A3 U A4 'We then define functions

on D. i 1, 2,3 Roughly speaking is the probability that

then e is an open path in 6T fl from to A1.3 separating z hour An; the

i emaining are defined similarly These functions generalize ctossin

probabilities: t ot: tuning to the 4-mat Iced domain DAo an open crossing

hour An to )13 is the same as a crossing of Da from An to A'3 separat

ing the point hom ,4,i; see Figure 5 Thus, the crossing probabilit:

T) is just the Mine of n(z) at the pointSinn nov moves a certain 'colour switching lemma' which implies an

equality between cert ain discrete derivatives of the functions r i; It will

follow that, as b 0, the functions f converge uniformly to harmonic

functions I' These functions satisfy bouuchu v conditions that ensurethat thec tr anshinin in a cc,intia math inviu iant way as D 3 is transformed,

7 2 Sin g nov Theatem

189

1

urine There is an open classing of I) loan A l to A 3 it and only if throeis an Open path how the ate A t to the arc = .1 3 U Ar that separates 111from A 2 'The indicated path separates R t ti am An and o how .4 2 , but doesnot separate a from .1 2 The function .1 .f(z) is the probability that there is anopen path in 67. fron t A t to si t: , sepa t ating z Crow .42

and that if D3 is au equilateral triangle with the cornets flnked, then

Bch t i is the linear function that takes the value 0 on the are .1; and 1on the opposite point. Cardy's formula in Carleson's form then follows

7.2-1 A consequence of an RSW-type theorem.

One of the key ingredients of Stith nov's proof is a consequence of theRusso-Sevmour-il relsh (11SW) Theorem. concerning open paths cross-

ing annuli with veiv different il/11C1 and outer radii So fat. we have

proved an RSW-type theorem only for bond percolation on the squarelattice Of the many /nook of this result:. most (perhaps all'?) can beadapted (ably easily to site percolation on the triangular lattice to

educe the following result

Theorem 3. .Let p> 1 be constant Them is a COP shin/ e(p) > Il such

that, if n > 2 and R pn by n rectangle In IR' with any orientation,

then the probability that the critical site percolation On T contogys an

open crossing ofR joining the two short sides is at least e(p).

Here. the notion of an open crossing is that km tmarlced discrete

domains. i e au open path uor,r vi in iL with m i . insicle

such that the line segments von t and et _ i VI meet opposite short sides

)1 R. For a detailed proof of Theorem 3 based on the strategy used

for bond percolation on 22 in Section 1 of Chapter 3, see BolloNis and

Riordan (2006b]

190 Conformal irraarian CC Smintatis Theorem

We shall use the following immediate consequence of Theorem 3 sev-eral times Let A be an annulus, i.e the region between two concentric

circles C1 and 0). We say that A has an open crossing in the lattice ST

if there is an open path from a site inside the inner circle C 1 to a siteoutside the outer circle C.)).

Lemma 4. Let. A be an annulus with ginner radius r_ and onto, radius

± If 7 +11 > 2 WO r_ > 10008, then the probability that A has anopen crossing in ST is at most (r_11 + )", where (11 > 0 is an absolute

constant_

Proof. Replacing a by a/2, we may assume that / 4. = 2k r_ for someinteger k > 1. In any annulus with inner and outer radii I and 21 . , wecan find six rectangles as shown in Figure 6. If 6/7 is small enough (and

Figure 6 Six rectangles inside an ru t /mins A of inner radius r and outer radius2r If each rectangle is crossed the long way by a closed path in OT, then theannulus A cannot be crossed by an open path

< 1/1000 will certainly do), then the shorter side of each rectangleis at least 26, and the 6-neighbourhood of each rectangle is contained inthe annulus, so the event that it has a closed crossing depends only onthe states of sites in the annulus As K(T) = 1/2, Theorem 3 appliesequally well to closed crossings Hence, for each of mu six rectangles,the probability that it contains a closed path joining the two short sidesis at least c, where c > 0 is an absolute constant By Harris's Lemma

7 2 Stair non's Theorem 191

(Lemma 3 of Chapter 2), the probability that all six rectangles containsuch paths is at least c6 Hence, with probability at least d i , there is aclosed cycle in ST separating the inner and outer circles of the annulus.

Recall that = 2 k t Thus, inside the given annulus A, we can

find k log(r+//_)/ log2 disjoint annuli A j C A with inner and outerradii and 2/7 , respectively Let E1 be the event that A; contains aclosed cycle separating its inside from its outside. If J_ > 10008, then

1P(E1 ) > di for each i. If A has an open crossing, then none of the eventscan hold. But the events E1 are independent. so

Pfil has an open crossing) < Pfno E i holds) = (E1) < (1 —,_r

and the result follows with a = — log(1 — c 6 )/(log 2) q

Thinking of open sites as black and closed sites as white, a path P ismonochromatic if all sites in P have the same colour. Lemma 4 appliesequally well to closed crossings, and hence to monochromatic crossings

Roughly speaking. Lemma 4 says that if we have a 'small' legion ofthe plane then, when our mesh a is fine enough, this region is unlikelyto he joined either by an open path or by a closed path to any part ofthe plane 'far away. To a certain extent, the form of the bound doesnot matter: any upper bound of the form f(r_/ r +) with 1(a) 0 as

0 suffices for the proof of Smitnov's Theorem. Note that even thisweaker form of the lemma is much stronger than the fact that there is nopercolation at the critical point:, which implies only that the pr oba.hilityabove tends to zero as I 4./5 oc with r_/8 fixed.

7.2.2 Discrete domains

The heart t of Sr MAT'S proof is a lemma stating that certain probabilitiesassociated to per colation on ST within a domain D are exactly equal (Ofcourse, this is just a statement concerning a subgraph of the triangularlattice T ) In presenting and proving this statement, we shall oftenconsider the (re-scaled) triangular lattice ST together with its dual, thehexagonal lattice 511 obtained by associating to each site v C ST aregular hexagon PL. in tire natural way, to obtain a tiling of the plane

By a discrete domain arab mesh S we mean a finite induced subgraphCs of ST such that the union of the (closed) hexagons 11„, v E

is simply connected When considering C,1 as a graph, the mesh 8 isirrelevant, so we may take 1 and view C5 as a subgraph C of T

192 Conlin nul l thew lance - Swirrow s Theorem

In terms of the lattice T On OTT the condition that C Gs) be

simpl y connected is equivalent to requiting that both G and its outer

bountho y T, 0÷ (C), ale connected subgraphs of T. w lane 0-(G)

consists of the set of sites of T\C adjacent to some site in G In fact, we

shall impose the Following additional restriction on out discrete domains

G: we shall assume that neither C not U+ (C) has a cat-vertex. This

restriction is it televant to the mechanics of the arguments that follow

but simplifies the presentation slightly

The inner boundary 1)-(0) of a discrete domain C C T is just the set

of sites of Cf that ate adjacent to some site of 7' \ C Om additional

assumptions ensure that both 0- (C) and 0 4-(C) are the vertex sets

of simple cycles in T: indeed. viewing G as a union of hexagons, its

topological boundar y OG is a simple cycle in the hexagonal lattice If

Following this cycle in an anticlockwise ditection, sm. the hexagons 0 e ,

sites of T) seen on the left hum 0- (C)„ and those on the tight form

O c (G); see Figure The condition that neithet G not if + (CI) has a

Figure 7 A discrete domain G (filled uncles and the lines joining thew) drawnwith the corresponding hexagons shaded The two thick lines are t he (desin T corresponding to t he Juliet and inner boundaries 0 + (G) and 3 (C) 01.61The topological boundaly OC of (the set of hexagons corresponding to) G is

I he cycle in /I separating shaded and unshaded hexagons

cut-ye t tex enstues that we do not visit the same ver tex of 0-(0) or or

eTf. (G) more than once 110111 now on we shall view 0-(G) and O H- (0)

as cycles in the graph 1

7 Billie1101,'S Theorem 193

In preptuation lot S i nn nov l s key lemma, we need a hu Him definition

A k-ranked discrete domain is a dismete domain G (ca Gs) together

with k distinct sites ,m; of its boundaly 0- (G), appealing in this()Mel as 0-(G) is crave/sec! anticlockwise. \Ve shall abuse notation bywriting simply G lot a hr-marked discrete domain (G;v 1 „ vb.) Palelyfor convenience, we inmose the additional condition that each marked

site v i is adjacent to at least two sites of T \G (This condition is verymild: any site n of 0-(G) that does not have two neighbours in T \G isadjacent to one that does.)

Given a k-marked discrete domai t we define the = sT(G) to bethe set of sites of it- (G) appearing between n i and in + , (with ok+ , =as 0-(G) is traversed anticlockwise We include both in and in.s. I into

In this discrete context, an open missing of C from A; to Ai issimply an open path in G star ting at a site of A; and ending at a siteof Aj

Given a kunat ked disci ete domain G = (G: (n; )). as ever y tutu ked sitein has two neighbours outside G, we can partition the outer boundaryi)± (G) of G into vertex-disjoint paths Ajij = Aj/j(0), 1 < i < k. so thateach AY- starts at a site adjacent to vi and ends at a site adjacent to cis.,Indeed. traversing id + (G ) anticlockwise we may take Ajijj to run horn thesecond neighbour of c i to the first neighbour of v i+j ; see Figure 8 Notethat a site r E C is adjacent to some site of Air if and only if v E

Recall horn Lemma 7 of Chapter 5 that a rhombus in the hit/rigida!'lattice alwa ys contains either a horizontal open crossing, or a 501ticalclosed crossing. but not both this statement,. and its moo!, extendsimmediately to -l-marked discrete domains Although the proof lo t thegeneral case contains nothing new. we give it in lull, since we shall soonuse similar ideas in a //101e complicated way

Lemma 5. Lel C be a lumoked discrete domain, and let = .11(G)1 i -1 11'haterer the stoles (t/ the sites in G. this graph contains

either an open crossing from A I to A 3 . Of u (lased ree`Otithe from

to As. but not Goth Ill pal (imam. the probability that C has on open

crossing Pow A, to .1 and the probability that C has tell open crosYng

from stt to As sane tO

Proof As above. let CY (CO denote the outer boundary of C. whichis partitioned into arcs Ajt Consider the partial tiling of the plane byhexagons, consisting of one hexagon for each site of C U (G). Colourthe hexagon ./1,. cot responding to c E C black if n is open, and white

•

•

•

•

•

191 Confunnel invariance chnirnoo's Theorem

•

•

ALVA A A AA A♦ V V

VA VAAA V

• •

•

Figure 8. A 1-marked discrete domain (68. e 2 . ea! et) (filled circles and linesjoining thew) The outer boundary eP(G), a cycle in T, consists of the hollowcircles and the lines joining them. The thick lines show the inner boundary0-(G) of C. a writer of arcs A. I < i sjj. shining endpoints and thecorresponding disjoint arcs C ac (CI)

if c is closed Colour the hexagons H,. corresponding to u E frt U At

black, and those corresponding to E AT U white, as in Figure 9

Let I be the interface graph for nwd by those edges of the hexagonal

lattice separating black and white hexagons, together with their end-

points as the vertices Then every vertex of I has degree two except

for loin vertices Th , 1 < i shown in Figure 9, which have degree 1

Orienting each edge of 1 so that the hexagon on its right is black, the

component of I starting at y i is thus a path P ending either at go or at

Suppose that .P ends at !Li , as tit Figure 9. Then the black hexagons

on the right of P form a connected subgraph of G U U At joining

:At to Al- . Any such subgraph contains a path within CI joining a site

adjacent to At- to a site tv adjacent to But then e E A, and

ar E A:3, so G has an open crossing horn A i to A:3. Similarly, if .P ends

at rp, then the white hexagons on the left of P give a closed (*.Tossing of

GI from 4, to Ai Crossings of both kinds cannot exist simultaneously,

as otherwise K5 could be drawn in the plane

The second statement follows immediately: as each site is open in-

dependently with probability 1/2, the probability of an open crossing

'7 2 Binh nav's Theorem 195

Figure 9 The hexagons Ii corresponding to G U &' (C). with those corre-sponding to the marked vertices / 4 , vo, ra, N labelled. A hexagon EL., v Eis coloured black for i = 3 and white for i 2,1 A hexagon H i ., v E C. isblack if t, is open and white if t n is closed There is an open path in GI nomA i to A 3 if and only if there is a black path in the figure horn A lt to .-11- andhence if and only if the interface between black and white hexagons joins y,to g.1

from A 2 to A., is I he same as the probability of a closed crossing fromAu to A.,

This lemma completes mu brief review of the basic proper ties of crit-ical site percolation on the triangular lattice T In the next subsectionwe present the first step in Sninnov's proof of conformal invariance

7.2.3 Colour switching

We are now ready to present, the key lemma in Smiruov's proof: this'colour switching' lemma states that the probabilities of certain eventsinvolving paths in a discrete domain are exactly equal. We shall shalloften identify a site of T or 8T with the corresponding hexagon Inparticular, in the figures that follow, rather than drawing site percolation

on the triangular lattice, we draw 'face percolation' on the hexagonallattice, since it is easier to shade hexagons than points

Let C be a 3-mar ked discrete domain, and let; x i , :to. .e, be three sites

in (3 forming the vertices of a triangle in C, labelled in anticlockwise

196 Confonnal im yal lance Stnirnov s Theorem

order around this ttiangle Thinking of open sites as black and closed

sites as white. we write 13 1 for the event that there is an open path

.joining x i to Ac = ATP). i e., an open path from x i to a site in .4i,

and 11'1 for the event that there is a closed path horn x i to Let

8 1 13,113 denote the event that there ate vertex disjoint paths Pi from

xi to with P i mid 172 open, and P3 closed; see Figure 10. Note that

Figure 10 1 Ire hexagons {do responding to a :S-marked discrete domain(G: c3). this time ‘‘ilhout its outer boundary II the Nei tices corre-

sponding to I he heavily shaded hexagons al e open and those correspondingto the unshaded hexagons are closed, t hen /3, /3.11 .3 holds

8 1 13Th;, = 13, q 13,} J IL, is just the box product of the events 13 1 . 13,

and II -3 . as defined in Chaplet 2 Define 8 1 11 .433 = L3 1 q II -Q C /33

sinalai b. and so MI

[he probability (HMI ibutimi associated to critical percolation on t he

iangultu lattice is of commit 'nese' \ Cd if we change the state of every

open site to closed. and vice versa Huts

Ft/3 1 11 -2 11 3 )=P(11,8 2 Ba), (5)

and so on. so there ate font potentiall y distinct probabilities associated

to events of the lot In X i 3",23 . X. ). E {13.111 SlIkh nov's Colony

Switching Lemma states tlmt three of these are equal

Lemma 6. Lela G be o 3-m a t/eed discrete domain and let c 1 . :c xa be

sites of G Jointing a triangle in G. labelled in anticlockwise rode) around

7:' Stith 1101' s Thew ein 197

this triangle. Then

P(/3 1 /32 11 .3 ) .111(D I -2 83 ) = POL L /32 B3 ) (6)

h t contrast to (5), there is no symmetry of the metal( setup thatimplies (6). Lemma 6 does not say that when looking for disjoint paths,the colours me it relevant as it makes no statement about IP(B, Bo B3)

To prove Lemma 6. we shall show that

P(B, Ilji/33 ) = P(8 1 Ijillja) (7)

Applying (5). or the relation P(B i 1 ,11'3 ) = 111-1 (8 1 which is essen-t ially equivalent to (7), one of the equalities in (6) Follows immediatelyThe older inequality follows similarly, or by relabelling

In tutu. (7) is equivalent to

IP(/3 1 11',D3= (8)

The idea of the proof is as follows.. Whenevet B 1 FI holds, one can find

l i n t el/mist' open and closed paths Q i and Q. , witnessing Bi

condition not only on B I ji. but also on the pietise values of the paths

Q i and Qo cart find these paths without examining the states of

sites -outside them

Next we shall show that if Fi l l ri B3 holds, then thole is nu open path

p, front 3i3 to .43 outside the inunermost paths Q j and Qo Ilms, the

conditional probabilih that 8 1 11:)/3: i holds is just the probability that

the -outside' contains an open path horn ,r 3 to A3 . As we have not yet

examined the states of alp sites 'outside' Q i and Q(i. this is the same as

the probabilit y that the domain 'outside' (2 1 and Q contains a closed

path from to A: 1 , which is the conditional probability that B111'41:3

holdsAt the risk of scorning too pedantic, we shall present a detailed pool

of Lemma 6 using the ideas above Note that a little caution is needed:on the level of the vague outline just given, it might seem that the same

argument shows that 1P(B 1 BoB3 B I M) = P(13 1 13,11 .1 / .8 1 13,0 In fact,

P(13, Vii ) and P(E3 i BoB3 ) ate not in genetal equal

To find the intim most paths witnessing B 1 111i, we shall fellow a return/

interlace between hexagons. Consider the /initial tiling of the plane byhexagons. with one hexagon ha each site of U (G) As holjae, we

colont a hexagon H, cm responding to a site v of CI black if e is open, and

white if v is closed This time. we colon y the hexagons corresponding

to Ali black, those corresponding to At white. and those to At grey

As before, let I be the subgl aph of the hexagonal lattice consisting of

198 Coriformal Smernov's Theorem

all edges between black and white hexagons, with then endpoints as thevertices This time, every vertex of I has degree 2 except for a vertexy where .41' meets At, and one or more vertices y i incident with grey

hexagons; see Figure 11. Let P he the component of I containing y.

Then P is a path starting at y and ending at a vertex :p i where hexagons

of all three colours meet.

Figure A 3-marked discrete domain with its outer boundary Hexagonscorresponding to 4 are black, those corresponding to At white, and those

to A. grey Internal hexagons are black if the cot tesponding site is open andwhite if it is dosed The interface path P starting at y ends at a grey hexagon

Let iv he the centioid of n,r0;r 3 , so a' is a vertex of the hexagonallattice Let Tr =emu; be the edge of the hexagonal lattice separating xrfrom tn, oriented towards tv; see Figure 12

We shall prove Lemma 6 via a sequence of three simple claims. In thestatements of these, the assumptions of Lemma 6 are to be understood.

Claim 7. If Br W., holds, then the Oiler face path P slat Hag at p tra-

vetses the edge e in the positive direction

P1'00.1 Suppose that the event B I IV, holds, and let P1 be any open

path horn 3$ 1 to A l , and P, any closed path from wo to ;19 Note that

PI and P, ate necessmils disjoint. As any site in is adjacent to a

7.2 Swinton . Theorem 199

Figure 12 The centroid w or a triangle ruts:es in 7', and the oriented edge-7 =71 of the hexagonal lattice H that separates x i from r

site in AI, there is a cycle C in the triangular lattice formed as follows:follow PI from x i to Ai Then follow (part of) At to its end Thenfollow par t of At, and, finally, follow P, from A., to :1,,} The cycle C' isshown by dotted lines in Figure 11 We may view C.', which is a cyclein the triangular lattice, as a simple closed curve in the plane in thenatural way

As we go around the cycle C' starting at: 1, 1 , we first visit black

hexagons, and then white hexagons Thus, exactly two edges of theinterface graph I cross C', the edge 7 shown in Figure 12, and an edgeyy'. These me the two edges of H shown with arrows in Figure 11. Thepath P starts with the edge gy', which takes it inside C' As all greyhexagons are outside C, the path P must leave C at some point, whichit can only do along the edge 7, proving the claim, q

Let P' be the path in the inter face graph I defined as follows: startingat y, continue along edges of I until either we traverse 7 in the positivedirection, or we reach a grey hexagon If P does traverse the edge 7 inthe positive direction, then P' is an initial segment of P Otherwise, P'is all of P. Let N(P'), the neighbourhood of P', be the set of sites of Gcorresponding to hexagons one or more of whose edges appears in P'

Claim 8. If P' ends unlit the edge then the set N(P') C G contains

(necessarily disjoint) paths Q i from. :f t to A i and (29 from, r, to Ait,

with Q i open and Qs closed.

Proof. The argument is as in the proof of Lemma 5: if P' ends with theedge 7, then the sites corresponding to the black hexagons on the left of

2(10 C'onforrnal Mem inner Stn0 non s Theorem

P' 161111 a e()Itilecticd subglaph S of At meeting A and containing

Any such subglaph .5' includes a suhgtaph 5'' of G containing both

and a site of C adjacent to At Every site u E .S' l is open: thecorresponding hexagon is black (as 11 E 5), and o E C. so u is open.

Finally. any a, E G adjacent to At is in A l , so there is an open path

(2, C C S C N(P) how c i to A, Similarly thew is a closed path

Qo C .1V(P) Flom to A, q

Together, the claims above show that BULL holds if and onl y if P'

ends with the edge "(7 The proofs of Claims 7 and S also given little

mote

Claim 0. The event 33 I II I,23 holds if and only if 13' ands with

and then, is on open pith P3 C C from 3 . 3 to A$ using no site of the

twig hbou t hood N(P') of P' /31 I I)IY3 holds if and only if P'

conk with and them k a closed path P3 C 0 from .r3 10 .4 using no

Sill! N(P')

Proof It static:es to move the first statement Suppose that B 1 11 433

holds, so then: are disjoint: paths P nom to A, with PI open Po

closed. atal fi , open Then the pool of Claim 7 shows that. tw ili t horn

its initial and final edges. P' lies entirel y within a cycle C' funned by PI,

Pi and parts of AI and (i e the dotted c ycle in Figure 11) Thus,

even site of AT/3”) is on of inside C' But 133 cannot moss C'. so P3 lies

entirely outside L. awl is disjoint from N(P')

The I everse implication is immediate hum Claim 8.

It is now pas) to deduce Lemma (i

Proof of Lemma 0 As noted eat Het . it suffices to show that (7) holds,

i e that

P(BI II ,Bt) =P(Bi

Let 9. he the state space consisting of all 2 1(I1 possible assignments of

states (open (a closed) to the sites of C3, and note that the probability

measure: P induces the nonnalized counting measure on U

For w E R. let P i (w) be the path I.' defined as above. with lespeet,

to the confignt ation w Let 2 he obtained nom w by flipping (changing

Croat open to closed of vice versa) the states of all sites in CI\ AI ( 331(w)).

The path latly be found stepdw-step. at each step examining the

colour of a hexagon adjacent to the cut tent path Hence. the event that

7.2 Stith not, s Theorem 201

p' takes a particular value is independent 01 the states of the sites of

G \ In patricidal P' (I) P'(w) so w" = w for any w E Q

Thus the map w' is a Injection

Suppose that w E B 1 11 -0.133 Then, by Claim 9, the configuration

Le contains au open path in G \ N(P(w)) from ,r 3 to A3 Hence ‘21

contains a closed path G \ N (1.3 ' (w)) = G \ tV(TIr2)) hot

Thus, by Claim 9, co' E B i lV)11 -3 Similarly, if w' E .T3 1 11011 ,-3 , thenu :c33a to A

./3,11 1,B3 . As w w' is a measure-preserving injection,

p(B 1 11 .083 ) = P(B, ll irlr3 ), completing the proof q

Let us remark that, while we could have used the Mtn/lace path P'

to refine 'howl most' open and closed paths Q 1 limn x i to A i and Q.,

front .1 2 to A ‘ r, there was no need: it was simpler to work directly frith

properties Of the interface itself

7.2,4 Separating probabilities

The next step is to give a kninal definition of the 'separating piobabil-ities: the limits of these probabilities will be harmonic functions on D

From this point On we shall view the discrete domains Go that we shall

consider as subgraphs of ST rather than T: this makes no difference tothe properties of Gs as a gtaph, but will be convenient for taking Bunts

laterLet Cs = (Gs: r:i) C ST he a 3-mat lord discrete domain .r‘ith

mesh 8, and let c E C be the centre of a triangle in ST As heline, we

may think of z as a vertex of the hexagonal lattice SH dual to (11 A

'fey idea in Sndinov's proof of Theorem 2 is to consider the probability

of the event

E?:(c) = { Cs contains an open A r- A, path separating c from At },

and the events E (z) and g(z) defined similarly, where, as before.

Ar i(C; s) is the path in the inner boundary of Cs starting at ri

and ending at or +1 , and At = Aj (Cs) C 0+ (C; (5 ) is the corresponding

path in the outer boundary of Cs Usually. we shall take z to be the

centre of a triangle in Gs, although the definition makes sense for points

z nearer to (or even outside) the boundary of Cs The subscript fi in

out notation is shorthand to indicate the dependence both On S and onthe discrete domain Gs.

Needless to say, by an open Ar -Ao path Pin C7,5 sve mean a path P

in the graph Gs C ST all of whose sites are open starting with a site

202 Conformal 11)0(11 Smini.ov' n; Theorem

in A i and ending with a site in An Such a path P separates z homAt if, when we complete P to a cycle C in (Yr using the arcs All andAt, the point z lies in the interior of the cycle C when C is viewed asa piecewise linear closed curve; see Figure 13 Equivalently, the path P

Figure 13 A discrete domain G's (circles and the lines joinil g thew) %viththe associated coloured hexagons: open sites ate shown by Ii led circles andshaded hexagons, closed sites by empty circles and unshaded hexagons. 'Theouter hexagons (those not containing circles) correspond to the u t ter boundary

0 11 (0s) of C,5 , divided into truce ales .4.11 , At and A4 at the thick lines Artopen A 1 --.19 path P and the associated cycle C separating z, the centre of atriangle in C, from At are shoran a thick lines

separates z front At if any path in 511 consisting of edges dual to bondsof starting at z arid ending on (a dual site adjacent to a site on) At,crosses a bond in P.

Note that P is requited to be a path, nor to revisit a vertex. Tints,in the case shown on the tight in Figure the event .Er (z) does not

holdAs before, let w E C be the centre of a triangle xr:ror t a in C. with

the vertices labelled in anticlockwise order, and let z be the site of 511

adjacent to a that is farthest from x 3 ; see Figure 12 Suppose that the

7 2 8711:177101t 's 'Theorem 203

Figure 11 The figure on the left shows a schematic drawing cif a 3-markeddiscrete domain a5 together with an A i -Aii pat h separating z hour A . If thesites on this path are open, then Ei,t(z) holds. The figure on the right shows aconnected set S meeting and Ao and separating z from A4 that does notcontain an Ar-Ao path separating z fron t At. If only the sites of S are open.theu Es(z) does not hold

event EPz)\E,(a,) holds, and let; P be any open path in C6 separating

horn At Completing P to a cycle C as above. the cycle C windsar ound z but not around w. Therefore, C' must use the edge t i rk 2 . Ifwe trace C by following P from Ag to A i , returning anticlockwise alongAt and At outside Co, then C winds i1/01/11(1 c ill tire positive sense, sowe trace the edge m i x, from fo to ,r 1 . Hence, P is the disjoint 1111i0/1 ofopen paths Pr from m to A i and A hem r2 to A,. As we shall nowsee, there is also a closed path P3 horn ,r3 to .43

Claim 10. Suppose that 3,,,,r.),:ra,lir and z ate as above (sec FiffittCS 12

and 15), and that the event g(z)\Ej(w) holds. Then B I M Va holds.e.., C, contains disjoint paths P, Mining ;c 1 to Ar = A(ac), with P,

and A open and Pa dosed

Proof As above. let P he an open A t –A, path in Cis separating z from

. We have already shown that P may be split into open paths Pi

horn 3,1 to A i and A from to Ag ft remains W find a. closed pathP3 joining ,r 3 to any such path is necessarily disjoint horn the openpaths PI and Pg Note that x:r itself is certainly closed: otherwise, theOpen path PI :r 3A separates 11., from

Once again, we follow an interface As usual, we consider the partialtiling of the plane by hexagons H,, corresponding to vertices v of Go U

204 Conformal 'now; iance Surirvoil5 Thcairtn

a+ (0 A l. where 0+ /Go) = U U is the onto bounda t v of 05We Cohan a hexagon H,. c E 61,r, black if l' is open and white if V

closed We colour H, black if e E At U and white if u E At; sc.;Figute 15.. Let. I be the oriented interface graph whose edges a t e the

Figure IS Ilse discrete domain (circles and the lines j0 n og them), witht he associated coloured hexagcms 'The inter lace I between black (dark andlightly shaded) and tc kite hexagons has exactly two vertices of degree I,namely yr and y2 An Open À - path P and the associated cycle C' sepa-rating z from At are shown by thick lines If :rr i is not connected to :L i bya closed pat I I , then t he while component containing ,r 2 is surrounded by aconnected set S of black hexagons: those not on 1" or At" U AT are shownlightly shaded The union of and P contains an A l -A 2 path sepal atingfrom Alt consisting of the clashed lines together with part of P The °limitededge 7 of the hexagonal lattice crossing Tow:, is indicated b y an arrow

edges of the hexagonal lattice wit h a black hexagon on the t ight and awhite one on the left This graph has exactly two vet tices of degreethe vett:ices .111 and rki shown in Figute 15

As x/i is open and is closed, the oriented int et lace I tont tuns the

7 2 Solo [mei ,: Theorem 205

edge J of tin hexagonal lattice (tossing .10.1 3 with 1 on the left: this

edge is indicated with an arrow in Figure IS

Suppose first that the component of I containing f is a path. Thenit ends at a vertex of I with degree 1. which must be it, But then thechine hexagons on the HI of this path Mira a connected set containing

3) . and meeting At As before„ it follows that there is a dosed path

ram ,e 3 to .21 3 , as required.

Suppose next that the component of I containing f is a cycle, as in

igme 15. Our aim is to deduce a contradiction. Let hr. 1m... h, be

the sequence of hexagons seen on the right of this ci cle as we trace itonce star ting hom f Then each // i is black. h i corresponds to 3.' 2 , h,

/responds to x i . and h i is adjacent to h i* , lot each i Iden tifying ahexagon with the corresponding vertex of the lattice, we do not have

= Mr an y i: otherwise. the interface cycle would visit w

twice

Recall that we lime open paths PI and P. , joining to A t and

to respectively. and that these paths can be closed to a rude C

sunounding c but not w by adding parts of At and A:. No edge of the

interface I can cross C. so ever v center h i lies on or outside C Let

he maximal subject to h, E P.) u At, and let 5 he minimal subject to

s > t and 11,, E Pr U At Note that 1 < r < s < as h i = :ro E Po and

h EPn

Let S = .11,„.I} Then S is a connected set of vertices of

the graph Cri — for rued from Cs by deleting the edge ..rTro: the

hexagons corresponding to S ate lightl y shaded in Figtue 15 As Scontains a neighbour of Pn U At and a neighborn of „Pi U At, it followsthat .PI U S contains a path F' in (1,5 — r i :r9 joining A i to A, (Thepart of tins path off P 1 UP) is shown by dashed hues in Figthe 15 ) The

corresponding hexagons are black (drawn with dark or light shading inFigure 15). so the path I" is open Fu Iv lies on or outside

C. so it separates z from At As P' does not use the edge x iitfollows that 13' separates w from At This contradicts mu assumptionthat g(w) does not hold q

The proof above is longer than one might: like, and can probable beexpressed more simply Note, howeve L that one cannot give a 'purely

topological' proof: in order to show that E, (m) holds, it is not enoughto find a connected black set meeting A i and A, and separating to frontAt Indeed, one might expect that the centre c of a triangle in ST isseparated fron t At by an open Am An path if and onl y if z lies 'above'

206 Conformal 11111(1rion.ye - Smirnov's ThCOlern

the unique path component of the black/white-interface in Figure 15,However, this is not the case, due to the possibility of the configurationon the right in Figure 1-1

It is easy to see that the converse of Claim 10 also holds

Claim 11. Let w E OFI be the centre of a triangle x i x 2 x3 in Gs, labelledin anticlockwise order If z E bf1 is the neighbour of w opposite x3 , thenE,(2)\.E,?(w) holds if and only if B 1 Hd173 holds

Proof The forward implication is exactly Claim 10. For the converse,suppose that /3 1 B911 13 holds, as in Figure 16 Then the disjoint open

Figure 16 A schematic picture of the event B 1 B211:3

paths Pi from to and horn 370 to An together give an openpath P flow A 1 to As P uses the edge x i :1H the path P separatesexactly one of z and w from At As there is a closed path from :r 3 to.43 , the point tv cannot be separated horn At by an open path ThusES(_) holds and Er, (v) does not. q

As before, given a 3-walked discrete domain (3,5 and a point E611, let Es(s) be the event that G5 contains an open A 1+1 --A i + 2 pathseparating z how At, where the subscripts are taken modulo 3 In thelight of Claim 11, Lemma 6 has the following immediate consequence

7.2 Swinton's Theorem 207

Lemma 12, Let ii:roxa be a triangle in a 3-malked discrete domain

Go, with its vertices labelled M anticlockwise outer. Let w E 811 be the

centre of the triangle, and let. zr, C2, Z3 E 611 be the neighbours of ay in

673. with z i and x i opposite for each i. Then

P(Eclit (z i ) \ E (Iv)) = P(ERz 2 ) \ ERiv)) = P(E,Nz3 ) \ E:J(w)).

Proof. Setting z z 3 . Claim 11 states that the events g(c 3 ) \ Es(w)and 13 5 B2 W3 coincide Permuting all subscr ipts cyclically, we have

E!(cr) \ = B2 B3 111[ and El(z2 )\ El(w) = 133 8 1 1 ,17o The con-

clusion thus follows horn Lemma 6 q

For the centre z E alI of a triangle in C,s, set;

1'n(z) = P(E,i5(z))

(9)

and, if to and , me the centres of adjacent triangles in C6, set:

h is(w, c) P(E:5 (z) \ .E7dtv))

Note that, trivially,

n(z)-n(rv)=iov,z)- (10)

As noted by Stith nov, there is a surmising amount of cancellation in

(10) It turns out that for z and Iv far from the boundary of L.), the

quantity ki5 (w,z) is of order 5 2/3 as 6 —4 0; the exponent 2/3 is the '3-

arm exponent' appear ing in relation (51) in the next section. In contrast.

fA(z) — is presumably of order 6. We shall not be concerned

with proving these statements, as they me not needed For the proof of

Srniinov's Theorem.

Roughly speaking, Lemma 12 implies that the discrete derivatives of

the a ate related to each other by rotation For a precise statement,

it is easier to work with integrals around contours. We consider only

discrete trianguleu contour s in G1,5, i.e , equilateral triangular contours

C whose corners are sites of ST and whose sides are parallel to edges

of ST, such that all sites on C me vertices of C. as in Figure 17. We

orient C anticlockwise Givers such a contour, for 1 < < 3 we define

the discrete coolant integral

I• D

) M C ) Clz

as the usual contour integral of the piecewise constant function f(z)

whose value at a point z of an edge 51, of ST on C is the value of

208 Conformal Moat lance - Stith not, s Theorem

Figure A discrete triangular contour C The d iscrete contour integral offi r wound C' is defined using the values et .111 at points of the hexagonal lattice

Along :rig. fen example. we integrate j',(ai)

IA(w) on the nearest vertex w of OH inside C, i e „ on tlw vertex of (if

immediately to the left of aril; see Figme 17 For example, if C is a

triangle with side length 6, mid a ' is the centte of the triangle C", then

f (z) .0u) identically on C. so 1n(-13 Inz)dz = 0

Let w = (—I + \/-3)/2 denote one of the cube toots of unity

Lemma 13. Let CI; be a discrete 3-hooked domain such that no point

of C is within disloace a of all three arcs of OGo. where a > 20006 If

C' is a discrete triangulat canton; in. Cs of length L. then

(1) (:)(1: — f,f(z) (lc < AL(61a)" ( 1 )

fm i 1.2.3 where Mc sumo set ipl is taken modulo 3, o is the constant

in Lemma 4. and A is an absolute constant

Proof Rix shall prove (11) lo t i = 1; (lir corresponding equations for

i = 2,3 follow by relabelling the domain

Let a' E OH lie the centre of a triangle in Cs, and let z be a neighbour

of iv in OH From Claim 10 above. if E,C(::)\E's(w) holds for some

then them are monochromatic (entirely open (a entirely closed) paths

from the three sites of Or adjacent to iv to the duce boundary arcs of

Cs The point w is at distance at least a (rout one boundary ale. .-1i,

say, so a monochromatic path horn a point adjacent to i v to As gives

an open or closed crossing of the annulus with centre ut and howl and

outer radii 6 and a. 13v Lemma the probability that such a crossing

7 2 Swirnov s Thcomm 209

exists is at most 2(10006/a)° = 0(010") Hence.

h is(w..z) = /a)"). (12)

uniforml y over C. w with a' the celiac of a triangle iu Cs and z st, w:

here and below, denotes adjacency in the graph

Let

tv)11,15(

wct c.- systs

where C' is the set of yet tices of OH interim of C. and z nutsover the duce neighbours of w in OH To evaluate — w we view w and

as complex munbetsFix w E C' , and let z i Ca be the neighbours of a' in anticloclavise

cadet, so

— w = w) = j (13!

From Lemma 12 we have

16(w, )= M(w..m) = Ow. za) (,A)

Also, applying the same lemma with the toles of :I , z pet nutted,

c . )) = Ca) = Z )

and

h(15(a)7 c3) = (

Setting z 4 CI , we thus have

(zi tv)q(ut, zj) — tv)1;;(v 1±i)j=1

w(zi -w)h,v(m.=i i i).i=1

3

(v)16(w

I=i

where the first equality is simpI N a relabelling Of the same sum. thesecond is Eton, (13), and the third (Li) and the following similarequalities Stumning over w E C"', it follows that

S2 = (13)

210 Conformal invariance - Snail noM ii Theorem

call write = — u01011, z) as 5'; +.57. where

5,, = (z- tv)11 16 (vi, z) + — z)h's(z, u,)it,,z€C"

and

(z — w)11:5 (ev, z).

The sum S7 has CALM) terms each bounded by Ssup„,,, , h[s(u,Using (12), it follows that

87 0(L(S I o)°) (16)

for eachFor z E C`) with iv ti z from (10) we have

( z — a)his(lut (a' Z )1/1,1(Z, w) = WX/116(111, h nS ( It ))= ( z — w )( — f4(w))

To obtain the must stun this last term over all unordered pairs {m, z}with m + z and E C' Collecting all terms of the Ram :17101"),

:17 E C. we thus have

= E E - Jig ?, )

E ('rv-z)f,R4v)- E E ( a ' — z)1,(10).wE C" z—w, ze4c,

In the first sum in the final line, the coefficient of each 1,((tv) is exactly0 There is one term in the final sum for every edge viz of SH crossingC from the inside to the outside. The edge ruz of 511 may be obtainedfrom the dual edge 1,"// E C of ST by a clockwise rotation through w/2followed by scaling by a factor 1/ \/[3 (For x, y and w, see Figure 17 )

Thus, z — w = ( -14/fr3)(y — :r) Hence, h our OM definition of thediscrete contour integral,

= (z — .0.1A(10 = — .11(Z) (1Z;

•(17)

VF3 c

As Si = + , the result follows from (15), (16) and (17)

Later, we shall show that if we have a sequence of 3-marked discretedomains Co with mesh S 0 that give finer and finer approximations

7 2 5'71th-two's Theorem 211

to a Jordan domain D3, then the functions Mz) converge to continu-ous functions f' on D Lemma 13 will imply that the (usual) contourintegrals of these functions around a given contour are related bynultiplication by co.

7,2,5 Approximating a continuous domain

In this subsection we show that one can approximate a Jordan domainD by suitable discrete domains without changing the crossing probabil-ity significantly, proving the technical Lenuna 14 below This result isimmediate for sufficiently 'nice' domains, such as rectangles. The readerAmested only in such domains may wish to skip the proof.Let us recall some of the definitions involved in Theorem 2. By a

classing of a 4-marked Jordan domain D4 in the lattice ST, we mean ajath in OT whose first and last edges cross the arcs A i (/). 1 ) and Aa (M),vith all vertices except the first and last inside D. If the sites of acrossing are open, then it is an open Glossing of D. 1 . As before, we writePo(D.,) = (D4 T) for the probability that D4 I as an open crossing.

The corresponding definitions for discrete domains are simpler: acrossing of a -1-marked discrete domain Gs is simply a path in the graphC:s joining A l (C; (5 ) to A3 (Cs) We write P5 (G. 6) for the probability thata discrete domain C,, C ST has an open crossing

Let us write dist(x, y) = - :y1 for the Euclidean distance betweentwo points :ET, y E R2 C. and (list (x, A) and dist (A, B) for the distancebetween a point and a compact set A, or between two compact setsA and B We avoid the more usual notation d(t, y) due to potentialconfusion with graph distance For two compact sets A, B C C, theirHausdoijd distance is

du (A, B) sup Udist(a , B) : 0 E A} u {dist(b, A) : b e

= int : A C B C A(=)},

where A V) denotes the (closed) e-neighbour hood of A. If = (P1))is a 4-marked „Jordan domain and (75 C 6T is a 4-marked discrete do-main, then Cs is E-close to Dt if the corresponding boundm y arcs of D.1and of G5 are within Hausdorti distance e, i.e if

driAi(G,5)) <5

fm < i < 4To understand the definition above, recall that .4 1 (.0.1 ) is an open

9 1 9 Conlin -mat invariance -- 5'ntit owl's Theorem

,Icadan cur ve in C In contrast, ttl i (a) ) is lo t wally a set of vertices of

G . i a set of points in (51' C C: condition (18) cm ' he hay' mete(with this definition. Often, however, we shall view Co as a union oclosed hexagons, so its boundary ,) is a piecewise linear curve in C.

hrthis case, A i (es) is naturally defined to be pat t of OCA) will IICVO

make a difference which definition we use: the two interpretations of:1;(96 ) give sets at Hausdorff distance (5/0 from each other, and thearguments that follow will not be sensitive to such small changes. Thus,:we shall feel fl ee to switch between these viewpoints without notice

Lemma 14. Let be a -1-marled Jordan domain. Then. some

do = 60 (D ) ) > 0, there are families {G' s ,0 < cl cz a0 } and 1. 0 ,T. 0 <rS < 6.0 of discrete domains C f517, with the following propertie s. As

cS 0, the domains Gs- art o(1)-close to D.1.

Ds( — o(1) P,s( D . 1 ) < 1?) (C ) + o( I), (19)

and, given -y> 0, there arc (5 1 , 11> 0 such that fur all r5 < S i , any two

points n: and of Cif with dist (w, z) < q may be joined by a path in GI

lying in the boll 13.,(w)

Iloughl n speaking, L.enuna 1-1 shows that. to in (WC ThCOlent 2, it suffices

to Ivor k with discrete domains Mote specifically, using the fact that (It-

is close to .D. 1 , we shall show that

Pi (Cn) — FID4),

which. together with (19), implies 'Theorem 2 The last condition in

Lemma If is a technicalit y that we shall need to ensure that certainfunctions go we shall define ate tutifin tid y equicontinuous as (5 varies

The rest of this subsection is devoted to the proof of Lemma Theconstruction of the domains (.7;t will be broken down into a se t ies ofsteps, and the proof that they have the required properties into a seriesof claims Before getting started, we present two simple facts aboutlot dart curves that we shall use

Lemma 15. Let D be a fixed Ionian domain with boundar y

s> them is an. = ) > such that if w, y E f and dist (.r. y)

then one of the two arcs into which and q divide F lies entirely

within the ball B (s')

Proof This is standard. immediate Iron the fact that 1 = (IT),

72 &nil noMs Theorem 213

v iarre i is a 1-to-1 continuous Mal) horn the circle T into C Such anap 7 is a homeomorphisur, so both 7 and its inverse ate continuous'unctions on a compact, set, and hence uniformly continuous

Alter natively, the fact that 7 is uniformly continuous implies that,given s > 0, there exist to = 0 < tr C • • • < < = 1 such that

len , 7([T, t. i .r. 1 1) lies in some ball of radius s./4 Any twosetsI F

with ] — i 0. ±1 modulo n are disjoint, and so separated by a positivedistance Hence, there is an r > 0 such that any two points y of Fwithin distance t lie on the same or an adjacent arcs F 1 . F ) . In eitherrise, them is an rue of F joining a' to y and lying within 8c(r). q

emma 16. Lc/ r 6c a Ionian chive boandinga domain D. and let.

E D be freed Given 5 > 0, there is an = g(F.C. ․ ) > o with the

following p i ppin ty far every point .r of 11, there is an .r' E D with

dist (r. .1 1 ) < c that may be joined to C by a piecewise intent path P with

dist(P, F) >

Proof Let as p}(.r i ) be a (minimal) finite set of balls covering r, andPick one point c t E 13,p)(x t ) fl D for each I, so every E F is withindistance 5 of some .-t t As D is a connected open set, each z i ma y beconnected to C by a piecewise linear path P1 in 1) The minimal distancebetween the disjoint compact. sets P Ui and C is strictly positive,so there is an q > 0 such that the 2q-neighbour hood of P is disjointhorn F q

One can show, that. given a ri-nun lied domain D., and an > 0, lotsufficiently small S there are 1-/ / ta/ ked discrete domains G:5 C STthat am s-close to D.:, such that any crossing of Gs in ST contains acrossing of D4, and any crossing of f).1 in ST contains a crossing of G,"ri-This implies that

P6(0,7 ) P6(D4 < Ps(Cr{ ) (20)

In fact, it will be cleaner to move a somewhat weaker statement, in-volving only crossings that do not pass to close to the /milked points Pi.

This weaker statement does not imply (20) Howevel, we shall show.using LC1/11 / 1a 4, that it does imply Lemma 1-1, which is strong enoughfon the proof of Theorem 2

The basic idea of the construction is as follows. If is a rectanglewith the cm ners un i tized, then we nay take (CT to be a slightl y !tinge'and thinner 'rectangle' in the lattice, and Crq to be a slighth shorter and

214 Conformal Invariance - no v's Theorem

fatter rectangle The case where D is a polygon is similarh easy Thegeneral case is not mate so easy: when we 'narrow' D in one directionwe may end up with a disconnected graph, for example. Also, when weextend it in the opposite direction, we may bump into ourselves, andend up with a domain that is not simply connected. These are not 'real'problems, but, nevertheless, it takes a fair amount of work to overcomethese difficulties.

For the rest of the section, let D. 1 = (Pi)) be a fixed 4-markedJordan domain, with boundary F. Given El > 0, in the constructiothat follows we shall choose other small quantities

> > > 5.i > Er, =

where ri.fi is a. function of so that 6; 4, 1 is much smaller than foteach i. In particular, we shall assume that 10005 i + 1 < say

Let s t > 0 be given Our first step is to simplify the crave F = OD mac.the nmrked points Pi As before, A i = A i (D) is the arc of the Jordan:curve F running from P. 1 to Pi± i . The arcs A t and A 3 are disjoint, andso separated by a positive distance The same applies to A 2 and Ai,:

Let Zo e D be fixed throughout, and note that dist(zo, A i ) > 0 for eachReducing E1, if necessary, we may assume that. dist(zo, > 106 t , and

that,

dist(A .4 3 ), dist(A2 , A. 1 ) > 106 1 . (21)

In particular. dist(Pi, Pi ) > 10s, for 1 0 with

E2 5 7-(r, Ei

where T(', s) is the function' in Lemma 15 Thus, if to and z are twopoints of F with dist (iv, c) < 69, then they are joined by an arc of F thatremains inside M, (w) Let Bi be the open ball of radius am around Pi:and let C1 be its boundary Taking the indices modulo 1, the al c A i ofF joins Bi to Bi .m, and so contains an arc F i disjoint from Bi U Bi+joining Ci to Ci+ 1 ; see Figure 18 Note that for j = i + 2, i +3, we haveclist(F i , pi ) > dist(A i ,A 1 + 2 ) > 105 1 . Hence, each arc F.1 is disjoint fromevery B1 . For each the set .4 1 \ F i consists of two arcs of F each ofwhich joins a point P1 to a point at distance 69 from Pj ; this arc is thuscontained in B_, (P 1 ) Thus, (F i , A i ) <

Let be the 'simplified' curve obtained by joining P i to Pi arid P1+1by straight line segments of length E, 0, as in Figure 19, and let D' be theinterior of I" Let he the arc of F' starting at Pi and circling at Pi-Fr,

7.2 Smilwov's Theorem 215

Figure 18 Th e joining to

Figure 19 'The 'simplified' curve r (thick lines), and its interim D'.. Thepoints P1 arc the cent res of the circles C; with radii am (drawn tvith solidlines) The dashed lines ate the arcs of F \ F', each of which remains

distance of some Pi , as indicated by the clotted circles..

and note that

di-dri , Ai) < (22)

for each iWe shall modify r in a way that is analogous to replacing a rectangle

by a slightly longer and thinner one The disjoint closed sets 1 0 such that

dist(r i ,F;) > 10E3(23)

for i

210 Cotd611nal en001 inure Smitnoe •; Theorem

Let

s;3/3)/2. (- I)

Where. as before, t is the function in Lemma 15 Let N 1 = ft' ) hethe closed s A meighboruhood of P i , and let acc (N1 ) denote its external

boomlarw, i.e tire boundary of the infinite component, of C \ N1 TheJordan curve er inds around F 1 . Flom (23), it does not meet:any other I .) , so it can cross r only hr the line segments inside Bi and

In tact, it can cross only the line segments L i and joiningto RI and Pi+t respectively Furthermore, as Fe meets Ci and (7.';+1at one point each. and lies outside Bi the curve (N1) meetseach of L i and exactly once, at the points Q; E and (2 1; E

With dist((2e,R) = dist(C4,Pj+i ) = E g - see Figure 20 Hence,D'(Nd consists of two arcs joining (2, and Cj ii . one inside r and theother outside r

Pignut 20 The curve 1 11 , (thick line) together 1A-ith its £ 4 -neighbourhood N1

(shaded) 'The extol nal boundary ir(Nd of N. , crosses r at two points. thepoints Q, CI, on the line segments L, and 14 The ca l ve 51 is one of thetwo arcs of (N,) joining Q, and Qf we take S, outside r for i 1,3. andinside for i= 2 4

For i = 1.3. let .55 he the arc of D=c (AC) outside F', and, a littledissonantly. for 2,4. let Si be the arc of D'(Ari ) inside r (Theexternal boundary of IV; = ft' ) is defined without reference to the restof F'. so Si is part of the external boundary of N 1 even when Si is

inside F'.) These choices for the arcs Si correspond to the operation ofreplacing a rectangle by a longer and thinner one, by moving the firstand third sides of the rectangle outwards a little. and the second andfourth sides inwards a little

We shall write Ri for the region bounded Iry the closed (nuke formed

7 2 SinirtunCs Theorem 217

by Si . . and the two line segments of length S.1 joining the endpointsof these curves

Claim 17. 11 7e can paramelf Ltd: the open Jordan carves Sm and I;continuous injections sm. gm : (0.1] C with sm(1)

i (0) E Ch E Ca + , and dist( sm !Mt)) < fin . alt t

Proof. Let tim(a) and i i (a) be palm/utilizations of the .lindan turves Smand I'm traced in the appropriate directions. Define a map a : I][0,1] as follows: fin each a E let Y„ be a point of I'm at distanceexactl y srr., hom X„ TOO E Such a point exists as Si is writ ofthe boundamy of Ni = j" t Define o(u) be !Ijm(o(u)), Note thati (0) = 0 and o(1) 1. as there is a unique closest point of to each

of Q i and (XThe straight line segment L„ = X„Y„ meets U only at its end-

points Also. the interior of L„ lies in A'm and hence in R i (As weCa avetse S. on one side we have the unbounded component of C \On the tithe/ side we have Ni and hence ) Thus, L„ separates Pi

into two pieces The boundaty of one of these contains all points ofSi appearing X„. and all points of L i appearing berme Y„: theboundary of the other all point's appearing after these points. As the linesegments L. L„, cannot moss, it follmts that o(ad) > o(u) lot > a.

So hut, we have found a win to hare Sm continuousl y so that a nearbypoint traces 1m monotonically. but not necessarily continuously We cantrace both curves continuously by 'waiting' on Si whenever the nearbypoint on I i jumps Mate fin i nally, wilting 0_(.) = sup{o(44): <and 0 + 40 = inf {o(P) : > .r}. as o : [0,1] [0,1] is increasing. wecan find conti nuous [Unctions a,(0 and a,(1.) horn [0.1] to [0.1] suchthat each a; is weakly increasing. amid

0.-(11/(0) � //2(0 5_ (4+(11(1))

lot even- a example, one can take u t (1) = sup{ : a + (A(d) <

and a 2 (1) = ni(t). Let = ;i(a2(1)) and g i(I) ai(w2(1)) Atpoints whew cr is continuous. we have la,(1)= 0(1 [ (I)), so, by definitionof o, the points ,s;(1) and ni (t) ate at distance exactly EA As both T.Cimand in ate continuous. and each discontinuity of a may be approachedFlom both sides by points at which o is contitotous, it fellows that thepoints ij i (o4(u l (t.))) are also at distance exactly 5.1 from si(t).

these points are within distance 2E,, of each Millet B y our choice (24)of :7:4 , it follows that the ate of 111 joining these two points lies within

9 18 Conlomat Mom-lance Surrnoo',4 Theorem

a ball of radius E3 /3 centred at either point:, and hence within a ball of,radius g 3/3 177.1 s3/2 centred at s i (t). As in (t) is a point of this atwe have (list (il i (t), -s,(f

So fa t , the functions v .) are only weakly increasing, so gi and arenot injections We can modify the tt ) slightly so that they are shied.,increasing, tot example by adding a small multiple of un to tt, and viceversa Using uniform continuity, this does not shift the points 1/1(1)s i (t) significantly, and the result; follows. 0

Let; P- be the closed curve formed by Sg, 8: 3 and S. together witl

the line segments joining the endpoints of these curves to the points pand let D" be the interior of 1---; see Figure 21

Figure 21 The curve F - (solid lines) and its interim IF The dashed li

ale the curves Each Si is palt of the external boundary of N i tr. It ll : hai = 1,3. S i is outside i 81 is inside r

Let IT denote the arc of P starting at P1 and ending at P1+1,so FT consists of Si together with two straight hue segments FromClaim 17, we can find corresponding parametrizaticms 1'; and FT thatremain within distance S3 fit particular,

for 1 < i 4. Also, piecing together these pain/netrizationsrye obtain

7 9 Son; non's Theorem 219

an t ett izations of the Jordan cm yes 17- and such that cotrespondingpoints ate within distance 5.3

Roughly speaking, we shall take CC to consist of all sites v of ST

vhose corresponding hexagons H„ a t e contained UnfoitunatelY,this set; need not be connected, so we shall have to be mot e careful

Suppose that

0 < 35 < Et/10) (26)

wLcre q is the (Unction appearing in Lemma 16, and let D (7 be the set

of sites v E ST such that H„ C D - Reducing 5, if necessary, we may

lusome that

< 05 4 /100, (27)

vhele 0 is the smallest of the angles at the cot nets Pi of 171Recall that. ca is a (fixed) point of D. with dist(co,r) > 10n, i.e with

(z0 ) C D Thom the construction of 11 ', we thus have Biie ,(zo) C

D' As r winds around zo, it follows that V also winds mound :9 , so

zi t) E in fact. B8 _, (cry) C D Hence, the hexagon containing cu iscontained in D-

Begat cling DS as an induced submaph of ST let Cloi be the component.of D:1 containing (the site corresponding to the hexagon containing) cu;see Figtne 22 We shall take 67 as one of our discrete apptoxitnations

Figure 22 A pint of C;;;" (viewed as a union of hexagons). together \in.( ' au-

other (small) component of the set of hexagons contained in D- 0:5 is

the component of containing the point; zu

to I/ To make G s- into a 1-marked discrete domain (Gis : 1.

we take vi to be the VCI ex of 0,1 closest to Pi

220 Confotwal )11-141 I )(Wet Sitthlt017 ti Theorem

Claim 18. Ire have d ji (OCC„F`) < si/10.

Proqf Suppose first that :r E 00 ,7 Then ,r is incident with hexagonsHe EC6: and H„, The hexagons and H„, ate adjacent:, and,H, E DI. so by definition of 0 161 , we must have H„. 0 D:.;" Thus, 11„

meets and dist (it, < 25 < £4/10Suppose next that :r E By (26) and Lemma 16, these is a point'.

x' E D- with dist(e,,r) < £ 4 /10 and a path P joining to co withdist(P,1 1 ) > 361 But then POs) C D- contains a path of hexagonsjoining zy to 3.!. so x' E G7 As :c 0 D D 07. the line segment ;ri/'meets 067, so dist.Cr, 0067 < .s.. / /10 q

Recall that ri. 1 < r < ate the ales into which the points P1 divideso [7- stints and ends with shot t line segments slatting and ending

at P1 and Pi _e j : the test of 17 is the onve Si As any point of Si is

w ithin / distance/ E. 1 of U. it follows how (23) that if .r E mid y E 17with i ) ate at distance dist(r.y) < 106, then, swapping .r and rj ifnecessa t y, we have j = the point :r is on the line segment of 17ending at Pi.+1 , and y is on the line segment of 17+ scatting at P1,1

As these line segments meet at au angle of at least 0. how (27) both e,and y ale within distance 105/0 < se/10 of Pi . i e well within the ball131

As is a component of the union of the set of hexagons containediu a siumb connected domain. it is simply connected Let us Imo/the bounda t v of 0 7: anticlockwise Each boundary vet tex v is withindistance 2S of a point of F. which must lie on some If o' is theboundary vet tex alto c, then a' is within distance 25 of a point of someIT from the remade above, if i j. then = {/.k± I}. and.1. y E B.,;,(124.), say hi ti the/ wonls. the closest boundary me r7 canonly switch when we ate very- close to some Pt . But in this legion. weknow exactly what r- looks like: it meets 13,„,„(Pk ) in two line segmentsFrom Claim 18, the boundaty of 0:11 comes within distance £ 4 /10 < £3of a Hence 00W meets B,„(Pk ) in a single ant, and, as we trace thebomicht l y of OCC, the closest curve r7 switches horn to r I whenwe trace this ate

Recall that c i is the closest vertex of 0 1s1 to the point P1 0 CIA:Thus. v, is a boundar y vertex of 07 lying i t / /3,00 (Pi ) Let .17 =

1 ,2 , v3 , ) be the bouudanv tuts into which the c 1 divide theboundan of QC.: Clain ' 18 and the comments Ame imply that

d it (Ar. [7) <.z. i fth (28)

'7 2 Stun nov s Them CM

for 1 < i < Using (25). it fellows that d li (AT F) < 2E 3 Hence.appealing to (22).

.11) < 2E 1(29)

abet words. C.;;; and D, 1 are 261-close

Claim 19. Let P he a crossing of 07. a path in 0:5" from Ai to

AT If P does not pass within distance si of any Pi , then P contains a

crossing of D Rom A t to A3 , and P inte i sects any crossing of D from• to A 4 that does not pass within distance ,;t of any Pi

Proof. Let ,r E AT and y E AT be the end-veilRees of P As :r E 0:7 the

point r! lies in D-, the intetiot of From (28), disti(x,r7) < e4/10

As :c is far horn Pi and Pi. it is close to a point of S i , and outside

D Recall that R i is the domain in the complex plane bounded by

and ri We have ,e E P t arid y E 133 . so P joins P i to R3 Let P' he

a minimal sub-path of P joining f i t to P3 Mien the first edge of P'

leaves R I , which it must do across ri and hence across F 1 C F. entering

D Similatly, the last edge of P' flosses 17 3 P can only cross F' on the

boundary of R i , f?3 , so the rest of P' lies inside D'. Far limn the comers

Pi , a point is inside D. i e, inside if and only if it is inside 17 Thus,

all other edges of P are inside D. and P' is a crossing of D from A I to

A3mw mussing of D horn A, to .1 4 retraining fat from all Pi

includes a crossing of 0,7 hour AT to AT which. by Lemma 5, mustmeet P q

Let 0;7 be defined as 0T. but with all subscripts cveled, so ice take89 and 84 outside F'. and 8 1 and 84 inside r

Claim 20. Let Di and E > 0 tie given IfE i >0 is chosen small enough.

mid CIW 07 Ulc constructed as above. then

Pitt() - E < (DI < &tact)+ E

Proof Opposite arcs of ate separated by some positive distancec > 0 horn (29), it follows that if Er is small enough, then any pairof opposite arcs of D4 , 0T or 0;1 are at distance at least c/2 FromLemma A, it follows that if El is small enough, then the probability thatam of these domains has an open crossing joining opposite arcs andpassing within distance E l of some Pi is at most s Assuming no such

999

Conformal invariance Smic no t t's Theorem

crossing exists. then by Claim 19 any crossing of G7 front Ai to A.(losses D, 1 front A i to Aa, so

R5 (D. 1 ) Ps(C; ) —

The second statement follows sit/Wally.

The final property that we require of out domains Gt is that nea t fps.points may be joined by a shot t path in the domain We prove this forGs; the conesponding statement for Gi ct then follows by per tnutit i g themarked points R.

Claim 21. Let y > 0 be given Their is an q = q(D4 , 1 ) > 0 with thfollowing property. If E l is chosen small enough, then any two points

w and z of C -5- with dist Z) < n OIC :joined by a path. iu G7 lying hrB., (w).

Proof We shall assume that ti < 7/2We may assume without loss of generality that m and z lie on the

boundary of Gs (viewed as a union of closed hexagons) To see this,consider the line segment wz If this lies in 67, we me done Othet wise,let x and g be the fist and last points of this segment on the boundaryof so 'on'. gz C G q Then disiff, y) < y < 772, so it suffices tojoins and y E OCC by a path in 67 lying within 13_ /2 (y). saw Thus,Claim 21 fat tv, z E (16- follows front the same claim fit iv', E

with 7 replaced by -I/2We may also assume that the line segment tvz does not meet OGri

again: other wise.. listing the points c1 in which this segment meets 00,7in order, it suffices to connect each to the next by a path in Thin(ri ) CB-, (w) Then the union of these paths contains a path in B-,(w) joiningay to

We shall choose 11(.0. 1 ,7) .50 that g < 7/100 and

5r/ < r (F. 7/3), (30)

where r(r ,E) is the function appeming iu Lemma 15 We shall chooseai <

If the line segment iv: lies inside Gs, then we ate done, so we mayassume that it lies outside As tvz meets the simple closed curve 8(77only at its endpoints, it divides \ GIs into two components, one of whichis bounded: see Figtue 23 Let R denote the bounded component. LetB be the at c. of 06T which. together with iv:, bounds R If B C L32(w),

0

7 2 Sunni Theorem 99:3

Figure 23 Pmt of CC (hexagons). including two points at and z on its bound-ary with dist(w, z) < 'Hie shaded legion R. is bounded by B C OCC, apiecewise linen/ curve joining w to z„ and the line segment The curvedline is part of F - ; the point :r' E is within distance 25 of the point x E DC",and B- C I? is the arc of from le ' to z'

then we ate clone, since B C joins ar to c Thus we may assume that

d ime is a point ,r E B wit h dist(t, > 7.

One of the hexagons meeting at say H1 , lies in U. while anothet,flo, does not, and so lies in 17; the hexagon trn is drawn with dashedhues in Figure 23 As Hi and Ho are adjacent, it follows from thedefinition of G ,7 that 17.- meets Ho. at a point x', say Let w' and z' bethe points at which we leave 1? when we trace the cm ye F- front a' inthe two possible directions, and let B- C 1? be the ate of E- horn Iv'

to z' containing x' As r- cannot cross 80:5-, the points t ri and z' lieor/ the line segment to:: Iu particulat , dist(uV , c') <

As noted above, it follows front Claim 17 that we may parametrize F -and so that co t responding points ale within distance .5. 3 < E t Romthe construction of f" (see Figure 19), it follows that we may parametrizeF- and r so that cot responding points ale within distance 2-7 1 < 2qLet w" and z" be the points of F cottesponding to the points ed andc' of r- Then dist(w". z") < 5q, so. IA' (30) and Lemma 15, one ofthe two arcs of E joining w" and z" lies in 13,)/3 (17" ) The points of I"-

Confot mal in vat iamt Suth6ov s 'Thcorem

conesponding te,/ this ate Mini an am B' joining er/' and lonminingwithin 1.3_,,,3±,:,(6 ") C do(w) This ate cannot be 13 - , which emit /tinsthe point: 2 r 13. 1 ,(r) Thus. B - U = [-

The calve B - U /IC:/ / is contained in B. and so does not wind moundcurve 13' U tv'z' is contained in B-,(w). and so does not wind

atoned r Hence. I3 U B' = P c does not wind mound x, cunt /Mid ing

E CriT CD q

The pool of Lemma 1/1 is now easily completed It remains to cuisine

that CI ,7! and (.7 -i" satisfy all the conditions in the definition of a discrete

domain The conditions that the domains and theit outer Imundmies

have no cut v e hex 11110 he et/Fenced by fit st teplacing tS bti et/10, and then

I mincing each hexagon by a 'hexagon of hexagons" As noted eat Mu the

condition that each marked vertex has at least two neighbours outside

the domain can be ensmed by moving each walked vertex at most one

step nronnd t he boundal v

Proof of Lent Illa .14 \Ve have shown that given a small enough Ei > 0,the construction tot described abm, e is valid lot all it < = E5(EI).

Considering a sequence of Values of m O. we may thus pick domainsCit lo t all rt smaller than some no in such a way that Cs is defined using

a value s- 1 (6) h m(rr) — 0 as el — 0.

The tcyuiretn ent that CST and /7 1 ale o(1}-close as — 0 Inflows nom

(29) Condition (19) is given In' Chihli 20, and the final condition of

Lemma 14 is gum/it/O'er! by Claim 21 q

Let Os I emai 1r that the proof of Lenuna 1-1 shows that when definingPi ( /7/, TL it does not inattel exactly how we neat the houndmvproof of SmirnoWs Them cm will be valid tot any definition of a c t ossing

of D., lin \vhich a flossing of the 'longer. thinner' domain C-:;" horn01 1 (LC ) to ../1/ ! (CC) t hat stays tar ham t he cot nets guru antees a crossingof D., how .ffl to 21 3 . and prevents a mussing of DA 110111 1 2 to .4 I . FOr

example, we could define a ("tossing of DA 111 ST from A i to .4 ) to beopen path in D Ma/ling and ending at points within distance 1051 of Arand .!1!

7.2,6 Completing the Proof of Smirnov's Theorem

Let D4 = (D: P). P./ . P i ) he a -1-ntai keel hot dan domain. rind as be-

rme let 7(D 1 ) denote the limiting ctossing ptolabilit S lot D I predicted

by Ca t ilv l s fonnula Thus 7(/). 1 ) is a cottfbnual MN/a/iron of D I which.

7.2 Smirnov's Theorem

225

following; Ca t lescm„ may he defined as follows: let be the unique con-k/rut/II map from to the equilateral triangle that maps , P2 and8 to the vertices (1.0). (1/2. fil/2) and (0.0), respectively, so /napsp, into a pout (r, 0). 0 < ar < 1 Then rc(Ds) =

Let C ST be the discrete domains approximating Ds whose exis-tence is guaranteed by Lemma 14 Note that there is a function E =

with s 0 as r5 -r 0, such that

0:sl. is ai-close to Ds (31)

Of coutse, the same holds for Cl,t, but, as we shall see.. it suffices toconsider G's-.

To prove Ibeotent we shall show that

Ps(G) rr(D. 1 ) (32)

as /5 0. where Ps(CCir ) is the pH/It/Wilily that the 4-marked discretedomain contains an open crossing joining its first and Hind boundaryat es. Indeed,

P,s(Gt ) r(D-1)

(33)

can be lamed in the same way as (32) and then P i (Dr) 7i(D4)

follows front (19). In fact. WI iting = P2 . P3, P I , PI ) lot the dual'marked domain. the consti net ions of the apptoxiination CI" to D4 andthe apt/it/xi/nation 61,1 to DI given in the previous section ate identicalHence. (33) follows hum (32). the /elation m(D. 1 )-+ 7,-(D1) = 1 (see (4))..

and Lemma 5 ha this leason we consider only C;"

Tot the moment, we shall regard the domains D., = (.D: Pt , P2. PI)

and 0,T (67: e t .r u:t vs) as 3-mallred, by fmgetting the fourthmailed point.. Pi of v., We write = .4 i (D// ), 1 < i < 3, fig theboundary mcs of the 3-tai Iced domain .D 3 = (D; P}, P4) \\'e usecorresponding notation 16/ the brt/Id/ay arcs of 67

Let f,l(z) he the 'et ossing-under' probabilities defined by (9), for thedomain Gs = = , 173 ) Thus, lot z the centre. of a trianglein CI . the have

C5(z) = P(E75(.:)).

where Es(z) is the event that these is an open path in 67 born =A 1+1 (67) to .:4; 4.2 separating z horn Ai (an arc of the outer bor/dittyof 67) Hoe, the subsoipts ale taken modulo 3

Let us extend to a continuous function /115 on the closure D of D

as follows Recall that is defined at/ yin ions points of i e..

226 Conformal Maw lance S tmirnov'a Theorem

the centres of various triangles (faces) of the lattice 81/' At the centreE 61-1 of any face of 45T meeting D. set: g,C(z) equal to the value

of JA at the closest point ay where /,( is defined. (H there is mote thanone such iv, choose one in an arbitrary way ) Note that, from (31), wehave

dist (II' < E + 26. (3

say\\1e extend hour the centres of the triangles meeting .D to the entire'

triangles as follows: first, take the value of fio at the corner :r of such atriangle to be (say) the average of the values at the centres of all trianglesincident with a and meeting D. Then divide each face of ST into t lacetriangles meeting at the centre, and define chi on each of these trianglesby linear interpolation Finally, 'forget' the values of 9i5 outside D toobtain a function with domain of definition D

\Ve shall Use two proper ties of the interpolating functions f./15.each !Ali is a continuous function on D Second, for any and c E D.there are points le t E 811 at which fA is defined. with 115 (w) < gfc(z)<

fRui f ). and dist (net„). dist (iv', z) < 2E.. where = 5(5) is as in (31) Thefirst property is inunediate from the definition of q The second followsProw (34)

Claim 22 The f (Indians MY' uniformly equiconlinnowi

Proof It suffices to show that t he functions fL ii ate tudIcainly equicout iu-

mmns Given ii > U, we roust show t hat t here is an q > 0 such that. for allto EDCC With dist (c. u . ) < q Mid all 8, Ale !Mee (c)— (415 001 <.1

ht doing this, we may choose an) 6 1 (ii) > 0 and restrict our atten-tion to the functions for S < 11 1 (0): whatever the values of (z)

> tSr (it) the linear inter polation in the definition of id ensures that thefunctions gA. e > 0r ( i3). are uuiforruhy ecpticont imams (indeed. indica rub,Lipschitz)

As no point of the 3-marked J o rdan domain3 D3 lies on all three arcs= (D3 ). there is a constant e > 0 such that

max dist (II', A i (D3 )) > (35)

fin aro. rc E 1)

Let us choose 'y > 0 so that < c/ltl and (63/c)' < d/2. where o >is the constant in Lemma -I By Lcuuua 1-1, if we choose Sr small enough.I Iwn t here is an I/ > tl such that any two 'mints g of GT. 8 < S I . at

7 :2 Sminuols Theorem 227

distance at most q are joined by a (geometric) path in G il that stayswithin tire ball 13s, CO Reducing 6 1 , if necessary. we may assume that

c(6) < for all 6 < 6 1 , where 6(5) is as iu (31)

Suppose that 6 < 6 1 . and ar z E D with dist(w,z) < ti/-f. It suf-

fices to show that g,15 ( z ) < 0,15 (w) +13 From the second property of

the interpolating fitnctions q listed above, there are ///, z' E 67 with

dist,(tu,a0 < ti/4 and dist(z, z') < 0/4 such that gA(tv) > ( to') and

9,i(z) < 0(z)). Note that dist(a",zn < q The points z' may be

joined by a path in CI; lying within 13-,(u') As (715 is 2-connected, it

follows that there is a path P' E 811 from w' to z' all of whose vertices

are the centres of triangles iu CT. with P' C B22(1/P1)Suppose that EA(z') \ EA(iii') holds Then there is an edge :ry of P'

such that EA (0\ EA (x) holds. But then, by Claim 10, the three sites

of 07 immediately next to :c are joined by monochromatic (i.e all sites

open, or all sites closed) pat hs to the three boundary arcs of CT Oneof these paths must: end at a distance at least c – – > c/2 front ni',say, where e is as in (35) Note that dist(r, w') <

We have shown that if EA (.-2')\ EA (tv') holds, then there is a monochro-

matic path crossing the anutilits centred at a/ with inner and outer radii

37 and c/2 By Lemma -1 and our choice of 2.. this event: has probability

at most

2 Hi

It follows that – (a'') < ci Thus.

< /Pc ' ) C ( 0 ' 1 ) + < +

as required q

The functions ifs take values in [0,..11 Hence, these Functions are uni-

formly equicontinuous and uniformly bounded Thus, by the Arzelft-Aston Theorem (see, for example, Bollobtis [1999, p. 90]), any sub-sequence of (g-, A, q) contains a subsequence that converges uniformly

to a limit IP), with each continuous

The last two (rather substantial) pieces of the jigsaw puzzle needed tocomplete the proof of Theorem are collected in the next two

claims The first describes some crucial proper ties of the possible limits(g 1 . 02 . ,q 3 )

Claim 23. Suppose, fa, sonic sequence -- 0, the G inks (11

228

C'onfot mat mew lance nov 5 l'hemmti

converge toilfu l oily to (g 1 // l1 ) avtk each f coalMoon Then or

any contom C in D. we have

g i+I (o) do = w g(c)do. (36)

an foi (my point o E Ai we ham

11'(o)= 0 and gi+1(z)-1- (0) = 1. (37),

who du 'opal 5C1 ipts are taken modals 3.

Proof Thioughout the proof WC consided onl y values of in the sequenceSS„. wilting /5 lot d„ to /noid nimbi/is/Atte notation

We stmt with (3(t): since the g i ale continuous functions on a com-pact set. they ate unikandv continuous Thus. it: suffices to considerequilateral triangular contours C with sides parallel to the bonds of T:an arbitrate contour C' in 11 call be approximated by a sum of suchCO/MOMS

Let C be an equilateral t tiangulin contour ill D with sides parallel tot he bonds of T Poi each J there is a discl et e triangular contain Cs inST tc id lin distance ci of C. Using (31). as C is contained ill t he open set,D. lot ri sufficient-1Y 5116111 we Iliac C C7 ,-c- Recall that the discretecontom chi? [ 15 (i:)d:: is defined IA int grating a function uhosc

/dues al given IA values of at lattice points wit hin dist miceand, at these lattice points. = rhii Since t he fittict ions ti,C a t e unifin nth-equicontinuous. it follo/ ‘ s hat

/)

dz !Edo) o(I.)

as S — whete the second it/leg/id is t he usual contom ioteguai Sincethe ri1c = q converge indlo t l i th to we have

( lc -r- = (I) f:/(:)(1 o(1)

as 61 = d„ — Combining the relations /Write with Lemma 13. wededuce that

0 g'(1) d o 0(1)

Since licit het integral depends on S. this proves (3(1)FO 1 (37). it suffices to prove the case i = 3. sad, flu a fixed o E

7 2 Smil noWs Theorem 229

Since the g at e continuous We mar assume that is not an endpoint

of .43,

disc(_. .*1r). (list (c.. A 2 ) > a (38)

for sotuw (I> 1)As 6 0. the 3-marked domain (3iis .s-close to D 3 with E = E(6) — 0

Hence. thew a t e points GITT with co e D and co — Shifting

1w at most 26. we mac assume that E 8H is the centre of a ttiangle

coo fro in C1sThus. f,((cs) is defined. and. as !LI — uniformly,

latch) fhis( c ) = g (c ) + o(1) (39)

Is — 0Flom (31) we may choose a porn ft's On the 'mutably of CT,T (viewed

as a union of hexagons) so that, its (5 —+ lb we have ws and hence

first (ms, co) 0 Let :I's E C',.,1" lie a site of the inner houndm v of Coll in

whose hexagon tux lies, so (list (,ro. ws) < 6 Note that it c = P l . then

we may take rs to be e 3 The :Hausdoi II distance between the discreteboundary arc A;(67) and the emit:imams air A i (Da) tends to zero

0 Thus. from (38).. lot 6 sufficiently small. the site lies on theate A3 of C1 ,7 (viewed as it 3-marked domain) and the point: ws on thecottesporaling ate of the bow/du t y 30W of the union 0 ,7 of hexagons

Suppose that D:51 (.7. 3 ) holds Thc‘n time is an open path limn Al(0,7)

to .'10(03l ) sepat ating limn At (Gill ), and, in pal t Rada', sepal atinghorn ws Such a path must moss the line segment: cows It follows thatsome site of within distance (list(:,, ws) + 26 = o(1) of c 3 is joinedby au open path to snow site at distance at least a — o(i) hum c3Lemma this event has mobabilitv o(11), so

=,i) = 111'(Eli3 (c 3 )) = o(1)

Using (39). it follows that cia (:) =Recall that 3! 3 is a lumndat V site of A 3 (0) within distance 6 of ms

Let Es be the event that some site in Baistfrc ?co+ t oc(zo) is joined by anopen path to A i (CI3- )U Aii,(CC) As above, LP(E3 ) = obi ) Let 0,15 be the4-marked domain obtained front the 3-marked domain 03 l by taking toas the 'Muth marled point Note that if c = so ,r 3 = el , then werecover the original -I-marked domain

If E3 does not hold. then no open path horn A t (gli ) to A; 1 (0) cantome within distance 26 of the line segment /tows. Hence E,i(z:s) holds if

and only if Cfs has an open (dossing h om A [ (Cs ) = (0;C: ) to .13(6"3):see figure 2-1

230 Conformal ar lance - Mirwou's Theorem

rs-- A!,

Figure 24 A 3-marked discrete domain 61 1,1 with boundary ales A i , 1 < i < 3,and the corresponding 4-marked domain 0,1j with boundary 8104 < i < 4,obtained by marking a point ass on .4 3 . If no open path joins A I U An to theline segment ,rozo, then an open path from .1 ' to A3 separates z,) from :At ifand only if it ends at a site of AC,

Hence,

P(Gs has an open classing limn A', to A'a ) = d(zs) + o(1) (-10)

Similarly. if E6 does not hold, then EA (As) holds if and only if C's has

an open crossing from ,41, to .4 ,',, so

has an open (flossing from A1, to A',1 ) = (zs) + o(1)

Using Lemma 5. it follows that

fd (z6) + ffi (-7. 6) = — o(1).

Appealing to (39) again. g 1 (z)+ 92 (z) = 1 follows, completing the proof

of the claim q

As before, + fr3)/2 be a. cube root of unity

Claim 24. Let A he the equilateral triangle with vertices n i = no = w

and va = w2 . There is a unique tnide (2 1 ,62 , y 3 ) of continuous functionsn.o D taking values in (0, and satisfying (36) and (37). MU theunore

y i (z) = (:z(z)), where h i is the lineal function On A with // 1 (z) equal

to two-thirds of the distance from z to the ith side of A, and y is the

unique conformal map from D to A whose continuous extension to F

maps Pi to ye for 1 < i < 3

Proof Let (0', r?, q3 ) be continuous functions on D satisfying (3(i) and

7 2 Smirh an's Theorem 231

(37), and set

Then g is continuous on D. For any= t +0_,92 w293

contour C in D we have

9 + g2 + w 2 g3 91 +w2 gl (a2.1 g'= 0.g =-

ice, by Morera's Theorem (see, e.g Bear don 11979, p 1661), the

function g is analytic in D.

On the b011./IdatY of D, condition (37) ensures that, fors E A i , the

value of g(z) is a convex combination of co 1+ r and44) 1+2 , i.e., that g mapsinto the line segment ig 4. 1 e i+0 Since g is continuous, it follows that

g maps Pi into Furthermore, as z traces the boundary of D in the

anticlockwise direction, g(z) remains within DA and winds exactly mice

around this boundary As noted by Bettina (2005], it follows that g is a

conformal map from D to A Indeed, applying the argument principle

(see, e.g., Beardon (1979, p 1270, the equation g(2) = w has a unique

solution for each w E A, and no solution for w outside A Since g maps

Pi to the craps g and y are identical

Arguing similarly, the function + g2 -1-:g3 is analytic in D. continuouson the boundary, and takes the constant value 1 on the boundary Thus

g2 + ga is identically 1 This means that the real-valued functionsg' are determined by g: for example,

(201 — g2 ga ) /3 + (g 1 + 9 2 + ifs)/3 = 211.e(g)/3 + 1/3

As g y, this shows that the triple (g I ,g2 ,g3 ) is uniquely determined.The triple given by g i (z) = hi(c,..-..:(z)) satisfies conditions (36) and (37),so the claim follows q

We now have all the pieces iu place to prove Sinn noy 's Theorem

Proof of Theorem 2 Let P3, P4 ) be a 4-mmked Jordandomain Let (.7:5- and G7 be defined as in Lemma 14, for all 0 < S < So,

where So > 0 is constant It suffices to show that Ps(C7) 7r(D4 ), sincethe same argument shows that Ps(G,I) ir(D4 ), and then .P,5(D4,T)g(D. 1 ) follows from (19).

Let f‘ and g,is be defined as above, using the discrete domains 67

We claim that /4; o y, where h i and y are defined as in (Hahn 24

Suppose, for a contradiction, that this is not the case, i.e that there is

an E. > 0 and a sequence (50 such that for each there are 1 < e < 3and 2 E D with

(41)�

232 Co Ilia ! Mal 1111 ,01 )(HUT S1011101 s Theorem

B y Claim 22. the functions g are unifounk eqificont Moons on the

compam set IT Since they ate also unifinink bounded the sequence

f ,, ,14 , ) has )1 1111ilin ink COMV/ gent subsequence Li t Claims 23

and 2[1, this subsequence CO/Wetgem to (h i 0 y.11 2 0 ;. // 1 o y) contradict-

ing (A )

The proof of Claim 23 slams that these are points c,[ n with

— Pr such that

P,;(67 ) (cs) o(1) = o( 1) 1 ) = 11 2 (y:(P. 1 )) o( )

Indeed. the first equation is exact Iv (-10) For c oo- PI : as noted Mat\ e. the

domain Cr[:, appeal ing in (-10) is exact IN t he -1-m anked domain 0:[[ in this

case As 11-' (y(1). 0) = 7(1 ) 1 ) by definition, t he proof is complete q

Site poi colatimi on the tiiiiiiLmtm lattice is t he mils standard pc[ i cola-

t ion model For g hick toilful/nal invariance is kiwi\ iv. tittle MC SOUR' 110/1-

standaid models to \\ inch either Sinn nov's Themem. or its pub! has

been adapted Indeed. Canria. Nelk111811 and SidthilVitillti [2002: 2004]

haVe ', 1 St ithiitiii0(1 t ith i lismal invariance lot certain dependent site peu b-

lat ion models nu the tiiaugular Lattice, obtained l, l imning a patIicnlur

deterministic cellular automaton horn an initial state given 11\ indepen-

dmo site percolation The\ establish continual invarance by showing

t hat the dependent model is hi a cethriu sense a small per tin bat ion' of

t he independent ittoth . 1 and t hen apply ing Themem

Chaves Lei 120061 defined a Lather unusual class of percolation

mids based on dependent bond percolation on t he I iangulal lattice.

mid muted the equiirdent of SmitnoCs theoreur t o t these models by

t I anslat ing StintSmituoc s pool to this context ft remains an blip)! ant

challenge to move conEumal invariance for am other standard model.

lo t example. lo t site or bond pet colation on the square lattice. or lot

Gilbert 's model or random \Mir : Hun percolation in the plane (see Chap-

ter SI

7.3 Critical exponents and Sell/ amin—Loewner evolution

Thee is a widel y held belief t hat the behaviour of various

phenomena'. inducting critical percolation. should be chat arm/ ized by

certain `ethical exponents" This opinion originated among theoretical

physicists hilt by WA\ mnuy mat hematicians hale been come t red Pot

percolation. these et itical exponents should (10 1 )0 1 1(1 on the dimension.

but nut on the details 01 the particular nrodel considered. T[his is a eels

7 3 C aritical exponents nd Schramnistocuinel evolution 233

substantial topic in its own right; a detailed discussion is beyond thescope of this book Hew. we shall briefly state the main rigorous resultsfor percolation: the existence and values of these critical exponents forsite percolation on the triangular lattice follow from Smitnov's Theo-rem and the work of ',alder. Schramm and Weiner. Our presentation isbased on that of Smirno y and Werner [2001]: we refer to the reader totheir paper for further details and full references

To describe the exponents associated to critical percolation we need afew definitions. \Ve shall consider only site percol ation on the triangularlattice, although the definitions make sense in a much broader contextWe write F1, 1, for the probability measure in Whiell each site ofthe triangular lattice I' is open with probability p, and the states ofthe sites are independent As before. we write Co for the open erratic;

of the origin, i the largest connected subgnaph of T containing (I. allof whose sites are open Fin the critical exponents. we use standardnotation: see, e g Reston [1987c]

Recall that the percolation probability 0(p) is defined as

= fril i ,(C la is infinite)

and that 0(p) = 0 it p < = 1/2. while 0(p) > 0 if p > 1/2It is believed that. ill a general set ting. the behaviour of 0(p) rrs p

tends to m hom above follmis a power i that

0(p) = (p p()4*()(1) p p, limn above. (42)

where ./.1 > (I is a constant that depends on the dimension but not thedetails of the petcolat ion model

Tinning to the size of the open cluster Co when it is finite. as earlier,we write V (p) = Ep (lCal) for the expected size of (number of sites in)Co As k(p) = x for p > mr . it is rather mote informative to modify thedefinition slightly.. and consider

\ f (P) = Ea ( lCallacokic) tP,,(IC01

so that \ f (p) = A(p) if p < pr Again, power-law behaviour is expected:

\ I ( p), _ as p— pc, (43)

where is a positive constant. and p p, bran either side.Fru theimme, at the critical probability. the tail of the distribution

23-1 Conformal hrouriunce Smintov'S Theorem

leol is expected to follow a power law:

(it < ro l < oc) = a-1/64-am as 11

as is the tail of the distribution of theradius 1-(C0):

(11. S r(Co) < oc.) = n- 1 / 6 , +00) as a

In the last definition, it makes no difference whether we measure theradius r(Co) of Co in the graph-theoretic or the geometric sense, sincethese two metrics are equivalent More precisely, we may take r (C0 ) tobe the maximum graph distance of a site at E C0 from 0, as before. (a wemay take r (Co) to he the maximum Euclidean distance of a site in Co

from the origin For site percolation on the triangular lattice, we have0(pr ) = 0. so the condition roj C Do can be omitted in (bl) and (15)

The ear relation length describes the 'typical radius' of art open cluster:writing for the Euclidean distance between rr E T C C and the origin,lei

1/2

5(M- -,01)),E7 1111 2 11',({0 — n rol < x})

It is conjectured that

= IP — Pc1 -1' for P (-1G)

The reason for the term 'correlation length' is that ti(p) is expected tobe closely related to the probability that two sites at a given distance1. arc in the same finite open cluster: roughly speaking, this probabilityshould decry exponentially with WE(p)

Finally, it is expected that, at the critical probability, we have

PP- ( 0 - (I) = i41 2-d-11+"(1) as (-IT)

where (/ is the dimension (so (I. = 2 for percolation on the triangularlattice )

The constants (.3„ 7, 6, (b., v and ri defined above are called criticalexponents, provided they exist We have used standard notation forthese exponents (though 6, is also written as p); the !bun of (47).example. shows that this notation is not the most natural for percolation.

In two dhnensions, it is not hard to deduce horn the Russo -Sevinottr-Welsh Theorem that it one of r l and 6, exists, then so does the other,and

d-2-hq=216, (48)

Y3 Critical exponents and .Schramm.-Loeumer evolution 235

'Aced, let x i , ,1], be two points at sonic (large) distance r II there is anopen path joining II and to, then each,r; must be joined by an open path

to the boundar v of the ball B, i3 Ur 0, say But with probability bounded

away from zero there arc open cycles in these two balls surrounding thecentres that ate joined to each other; relation (-IS) their follows usingIan is's Lemma (Lemma 3 of Chapter 2).

Mester' [19871; 1987c] established highly non-trivial relationships be-men the various exponents for two-dimensional percolation. Firstly,

building on Iris work on the 'incipient infinite cluster (Kesten (19861),he showed that if t i exists (or, equivalently. it exists), then so does 6',

with 11 (5 ± 1) = Fur then/lore, Ire showed that if the exponents ri and

S exist, then so do the other exponents, and

2 1 / 6— IIj

6-I- 1 ' + 1sind " ± I

=

In pa/ for two-ditnensim al percolation the 'scaling relatio ns'

+ 2 ,3 = 0(5 ) and = ti(2 1/ ), (J19)

hold, as do the '111Telsealing relations.

—dO, = + I and 2 q I (50)

S +

It. is believed that (-19) holds in all dimensions, and that (SO) holds fo r

d < 6: see Chimmett [1999] A yet y° large number of papers have beenwritten about the ethical exponents associated to percolation aud therelationships between them: see, for example, Rudd and Frisch [1970],Wu [1978], Kesten [1981], Aizemnan and Newman [198-1], Durrett [1985],Chayes and Chayes [1987], lasaki 11984 Mester and Zhang [1987],IKesten [1987a: 1988] and Hammond [2005] Burgs, Ch ryes, Kestim andSpencer [1999] proved the deep result that the hyperscaling relations(50) hold as long as two assumptions are satisfied: 5, exists, and, atp = N. the crossing probabilities for cuboids with fixed aspect ratios arebounded away from I. (Their results are stated for bowl percolationin Za , but proved in a more general setting.) The latter assumption isexpected to hold for cl 6

Bettuning to two dimensions, Selo anun [2000] studied a certain scal-ing limit of 'loop-erased random walks' in the plane, which we shall riotdefine. He defined a family of random curves iu a domain in the plane,whose distribution depends on a real parameter which he called thestocli astic Locivrod evolution with parameter h, and denoted SLE„.. The

230 Confirm& imam iance 5'mirnotc.s Theorem

random cunt S L E„ is often known b y the name Schwalm Locione i con:lotion. SatWI WI showed that if. as conjectured. the scaling limit of theloop-erased random walk is confounall y invariant. then it roust be SLE),Fun thermore. he showed that SLE6 is the only possible cordon wally in_variant 'scaling limit: of el itical percolation on a lattice in a sense that

we shall not untke preciseStith nov [2001a] proved that critical site percolation on T does indeed

have a scaling limit. and that this limit is conformalle invarinrut and thusequal to „SLEn Considering the face percolation on the hexagonal latticecorresponding to site percolation on the triangular lattice. it liAlows

that. for p 1/2. the long-range behaviour (i.e . limiting helm ion ic asthe lattice spacing tends to zero) of interfaces between open and closed(black and white) regions converges to SLE1

In a series of papers Lawler. Schramm r i nd \Ve t net [2001c: 2001d;20026; 200221] studied the behaviour of SLE: ire particular. the y deterironed Nal •cl it ical exponents' associated to SLEi; Combining theseresults whin those of Slid/ /10V [200!a]. SLIM mixt and \Vet nett [2001] estab-lished the existence and values of the various critical exponents fin sit e

percolation on T

Theorem 25. Poi site parrolation on the biangulat lattice the critical

opponents 3. -; ti and q di/toed implicitly by(42). (-13). (.16) and (.17)

and flu' inners

15;

5 -13 4 5= and= TT-

These values for the critical exponents coincide with the predictionsof theoretical physicists: see Kesten (1987c]. Sion 110V and \Victim' [20011and the references therein As noted above. 6, = 1112 = 5/48 followsrelatively easily, and d = 91/5 161 IOWS ftun a the results of [(estctu [1984

The proof of Theorem 25 is based on crossings of annuli lb sa y afew words about this proof. denote tu ALt the event that in the sitepercolation on r restricted to the disc /3 R (0), there are (at least)distinct open clusters each of which connects a site in B, (0) to a sitenear the boundary of BE (0) For fixed i. the asymptotic behaviourof P IT:, (A; ??) does not depend on as long as n is large enough fortins probability to be positive \\:n it ing A llt for rrü, ??. say, S i nn no\ and

\\hiller noted that. by the results of Kesten 119874 Theorem 2,5 fellowshorn the tur, relations

P1 / 2 ( A IR) =

(51)

0

.1 ideal cdpoilelds and Schramm doewnci cuolution 237

wd

P l/ q (-1 ;?) = R-5/ (52)

Clearl y. the [list of these total ions states siniph that, exists and takesthe value 5/iIS

Lawler, Schlanun and \ Vet net (20021)] showed that relation (51) Eel-

haws flow SItlit 410V ' S COltrOl inal 'mini lance tesults and then results onSLE6 Smiinov and Wei net (20011 then punted (52) and thus 'Theo-

em 25In fact_ Smir uov and \Venni' moved a none genital statement about

ct ossings of the tumulus A); 1?) emitted tit the origin. with inner andante! tadii / and R respectively As helmet we think of open Pathsas black and closed paths as white Given a sequence c = (c;)1=! E

,If of colones. let H,( 1 , I?) he the event that A(t. I?) contains j

vertex-disjoint monochr intuit ic paths P i whew P1 has cutout et,

each 1); starts at a site of .4(7 1?) adjacent to the inside of the annulusar id ends tit a site adjacent to the outside. and the initial sites c i al the

P1 appeal in the cyclic oi del c((, , v 1 around the cliche of tachus(As the paths do not (loss.. then final vertices apnea/ in the sane cyclic

order al ound the outer tit cle ) Let Ge (1. I?) he the emus corr esponding

to 1-1(1 17). hut defined in the hull'-annulus

= E C : t < < R l ite(z) >

It is not haul to sett that. For crit ical site percolation on It the proba-bilit y of the event C,(1 .1?) does not depend on the terms of the sequencec, only on its length Indeed.. the woof of Lemma 5 allows us to deli ie a'lowest' (clockwise-most) open missing of e."(;.1?) I?) From the inner tildeto the ante! citele. 'whi t /revel such a crossing exists Having Found sucha classing. Pi , we may then look lot a lowest open at closed crossingabove PI in the same wa y. The lowest classing P i may be found with-out examining the states of sites above Pi . so when we look fin the nextclossing, the inobabilitv of success does not depend on whetIon it is tinopen cat a closed crossing that we seek It follows sin/HaiIv that

^us(C',(; R)) = R)

lot sonic (ti (t, lit ) that depends on the length of c but not the actualsequence Sinn ion and Wet nei (2(101 showed that. , it j > 1 is fixed andd is huge enough. then

(53)(1 ) (t .1?), 1?

238 Conic)! Mal inner iance on 's Theolvin

as R xn This exponent JO + 1)/6 is known as the j-arnt exponezin the half-plane

Returning to the full annulus, one can show (see Aizenman, Duplantiétand Allat011y 119991) that P ir, (fIc ( t, R)) is independent of the coloursin the sequence c, provided C contains at least one B and at least One

H' This is related to Lemma 6 above: if the sequence contains twoterrns of opposite colours, then it contains two consecutive terrns, andone can start by searching for an Innermost' pair of paths of oppositecolours Then, working outwards, each remaining path may be found asthe lowest crossing of a certain region, and the probability of finding thenext path does not depend on the colour

\VI iting bj (t, H.) for P i/2 (He (t,R)), wirer e c is airy sequence of length jcontaining at least one B and at least one H', Snriu nov and Werner [2001]showed that if ) > 2 is fixed and r is large enough, then

Di (r,B) RH12- 0/12+00)

(54)

as R ce This exponent (12 — 1)/12 is tire (inallichromati i) 'harm

exponent in the plane The values of these exponents, and of the half-plane exponents, were pr edicted correctly by ph y sicists: see, for example,Salem and Duplantior [1981], Aizennum, Duplantier and Akaror/y[4999]and the references therein.

If one is careful with the exact definitions at the boundary (ratherthan glossing over them as we do here in our brief description of theresults), then the events .4 /;:i? and fle(r,R) coincide, where c is thealternating sequence of length 2k: the k black paths correspond to thek open clusters joining the inner and outer circles of A(r,/?), and tirewhite paths witness the fact that: these clusters are disjoint. Hence, thecase 4 of (54) is exactly (52)

\\T ifton': going into the details, we shall sty a few words about theproofs of (53) and (54), and hence of Theorem 25 As noted ear lie, t hekey elements are the result of SMitlIOV12001a1 that the (suitably defined)scaling limit of an interface in critical site percolation on T exists and isequal to SLEG , and the results of Lawler, Schramm and Werner on thebehaviour of SLE6 Putting these ingredients together, one can showthat

a d ( i . R) (R 11)--.1(i+1)/ti+o(I)

(55)

as R, r — x with .1711 fixed. (In fact, to obtain (55), one first needs an 'apriori' bound of the Ram a 3 (1 R) = 0(17- I —I ) as R cc, for constantsI and > 0; see Smirnov and \Verner 120011 and the references therein )

7 3 Critical exponents and Seltrantra-Locumei evolution 239

To obtain (53), one also needs 'approximate multiplicativity' that, for< 1 < 13,

(1,1(1 1, 1 -2) 65(T2, 1 3) e(a10r,13));

see Kesten [1987c] and Kesten, SidO/aViellIS and Zliang [1998] WhenConsidering paths of the same colour, this relation is fairly easy to derivefront the Russo-Seyrnom-Welsh Theorem Fortunately, for half-annuli,One can assume that all paths have the same colour: The correspondingrelation for hi is much harder; see SatittION" and Werner [2001]

Finally, let us note that, in the annulus, the restriction that not allpaths be the same colour really does seem to matter It is likely that

1Pr/2(lf,(7,R))

a, cc with t and j fixed, where c is a sequence of length j with =.13 for every i However, these 'monochromatic' exponents 7; are verylikely different horn the multichromatic exponents (j2 - 1)/12 above

"rm. ticulm, Grassberger [1999] reported numerical evidence that =0.3568 ± 0.0008. The numerical value, or even the existence, of -;; forj > 2 is not known, although Lawler, Schramm and Werner [20026]showed that 7,, exists and is equal to the maximum eigenvalue of acertain differential operator

Critical percolation is not the only discrete object known to have acargo, mally invariant scaling limit described by SLE: Lawler, SCl/181M/1

and Werner [2004] have shown that one may define certain natural scal-ing limits of loop-erased random walks in the plane and of uniformlychosen spanning trees in a planar lattice; these are related to SLE2 andto SLE8 , respectively.

The brief remarks hr this section hardly scratch the surface of thethe theory that has grown out of conformal invariance and the study ofSLE For a selection of related results see, for example, Schramm 120014Lawler Schramm and Werner [2001a; 2001b; 2002a; 2002c; 2003], Klebanand Zagier [2003], Beflara [2004], Dubedat [2004], Than [2001], Morrowand Zhang [2005], and Rohde and Schramm [2005], Informative surv eys

of the field have been written by Schramm ' [2001b], Lawler [2001; 2005],Werner [2004; 2005], Kager and Nienhuis [2004] and Canty [2005].

8

Continliuni percolation

Shortly after Broadbent and Hanttnetsley slatted percolation theory rind

Eras and Bányi 11960; 1961a1, together with Gilbert 119591, founded,

the theory of random graphs. Gilbert. 119611 started a closely related

area that is now known as continuum percolation The basic objects of

study are IV/1(101U g cometric graphs, both finite and infinite Such graphs

model, for example a network of transceivers scattered at random in the

plane or a planar domain, each of which can communicate with those,

others within a fixed distance

Although this field has attracted considerably less attention than per-

colation theory, its Minot tante is undeniable: in this single chapter, we

cannot. do justice to these topics. Indeed, this area has been treated

in hundreds of impels and several monographs, including Hall [1988]

on coverage processes Monts 1199-11 on random Vorolmi tessellations,

{tester and ROY [1996] on continuum percolation, and Penrose 120031

on 1. rtmlotn geometric graphs These topics are also touched upon in

the books by Mathison [1975]. Santal6 119701, Stoyan, Kendall and

Mecke 11987: 19951. And kutzutnian [1990] and Molchanoy [2005]

In the first section we present the most basic model of continuum

percolation, the Gilbert disc model or Boolean model, and give sonic

fundamental results on it, including bounds on the critical area In the

second section we take a brief look at finite random geometric graphs,

with emphasis on their connectedness The most important part of the

chapter is the third section, in which we shall sketch a proof of the

analogue of the Hatiis Kesten result for continuum percolation: the

critical probability for random Volonoi percolation in the plane is 1/2.

lyre shall frequently circuit/1ns sequences of events (A„) with E(.4„)

as n :xy, using standard shorthand, we say that A„ holds why or

with high probability in this case. As usual, an event holds almost snafu,or as . if it has mobabilibt

8 The Gilbert disc model 2-11

8.1 The Gilbert disc model

for r > 0. the yerfes set of the standmd Gilbert diNC model, or theBoolean model. G,, is a set of points distributed 'uniformly' in the plane,with density 1 To obtain G,, join two points by an edge if the distancebetween them is at most i The trouble with this 'definition' is that itis not clear how we can choose points unifinntly, with a certain densityn fact, it is easy to turn this hopelessly loose idea into a definition

of a Poisson process in the plane, the 'proper' way of selecting pointsuniformly

Let A be a positive real number.. and let PA C be a random count-ably infinite set of points in the plane Let us write bi A (U) fin thenumber of points of PA in a bounded Bore! set U: note that i t A (U) is arandom variable \Nje call PA a homogeneous Poisson process of intensity

(density) A it the following two conditions hold

(i) If U,. ..U„ are pair wise disjoint bounded Borel sets, then therandom variables p A (Uj j ), pA(U„) are independent;

(ii) For every bounded Borel set U, the random variable / A (U) is aPoisson random variable with mean AlUi, where IUD is the stan-dard (Lebesgne) measure of U

It is easily seen that there is at most one random point process (arandom countably infinite subset of the plane) satisfying these condi-tions; in fact, as pointed out In fiónyi in the 1050s, condition (ii) alonedefines the Poisson process PA In the other direction it is not hard toshow that !hele is a point process satisfying conditions (i) and (ii): forexample, one can use the following concrete construction

For A > 0. let {Ne i : j) E 2'} be independent Poisson randomvariables, each with mean A Thus,

P ( X i ,i = h ) = (CA\k/,•!

for k = 0. 1

(I, )) ELet be the unit square with bottom left vertex

Q;; = {(i.!/): I ti .< i 1. j < y < p± 1

every j) E E.2 , select Nu points independently and unifinudyfrom then the union of all these sets has proper ties (i) and (ii), sowe may take this as the definition of PA

Although we shall not study it here, let us remar k in passing that aPoisson process •Pf of intensit ysatisfies (i) and (ii), except that the

2-12 Continuum prErrolation

mean of µj(U) is given by 1 1 , /, where is a lion-negative _el:

integrable function

RenitIllilg to Pa, note that if Z is a Bore! set of measure 0, then the

probability that PA lets any point in Z is 0; hence, in what follows, we

shall assume (lint PA fl Z = th lot all measure 0 sets Z that we consider.

For example, we shall assume that no point of PA is on a given (fixed)

polygon; a little wore generally, given a plane lattice, every point of PA

will be assumed to be an interior point of a face

For A > (1 and 7' > 0, let a, a be tine random geometric graph whose

vertex set is P. with an edge joining two points of Pa if they are at

distance at most 1; see Figure I We call C, A the Chlber t model with

parameter s r and A, of the Boolean inadel with parameter 5 r and A. With

this notation, the standard Gilbert (or Boolean) model is C. = C

Figure I. Pail of Hie graph C a (dots and lines) Two vertices ale joinedthey are within distance i e , if each lies in the shaded circle centred on theother

The model GC , A has numerous variants and extensions First, let

us rescale by writing H, ,A for C.), A. This resealing is not entirely

pointless, since it allows us to define a random subset of R2 in a natural

way, by writing D = .D, ,A (P A ) for the union of the discs of radius 7'

about the points of Pa; see Figure 2 Note that the probability that

there are two points of out Poisson process at distance exactly r is (l,so it makes no difference whether we take open discs or closed discsThere is a. one-to-one cot respondence between the compor ter its of C2,. a =

(1 , ,, ',CPA ) and those of D, A = D, a(PA ); in particular, as PA has (a s )

8.1 The Gilbert disc model 243

Figure 2 Pail of the jit atilt G2, A (dots and lines) The shaded region is D,. A.Two VW tices of (.32,. A toe adjacent if and only if the conesponding shadeddiscs meet

no accumulation points, Go t A has an infinite component: if and only it1),. A C R2 has an unbounded component Note that although D, A isveto close to H, A. the two models are not isornol pine, since the discof radius I about a point may be covered by other discs making upD„ A The advantage of consider ing rather than Go, A is that thecomplement: of .D, A, the 'empty (or vacant) space' 1:3, ,A = 1R 2 D, A not

coveted by the discs, is just as interesting as the original set D,.A

particular, we can study the the component: structure Of E„ A as well:we can look for an unbounded vacant component, i e , an unboundedcomponent of E,,A

We may define variants of the graphs (1, ,A and 11, A, and sets D, ,A andE, A, by replacing the circular disc by an arbitrary centrally symmetricsubset of R2 . Thus, given au open symmetric set A C R 2 for 11 C p2

we write G A (11 7 ) for the graph with vertex set 11 1 hi which two verticesx and y ate joined if x—y E A. Equivalently, for x E R2 , set Br B+x,where B = and join two points x and y of IV if 13, n B = Again,the union jrciv B„ reflects the structure of G .4 (11') If A is the discof radius r, then we may write G, for GA (IF) In two of the mostnatural variants A is taken to be a square and an annulus Of course,the model also generalizes to d dimensions in a natural way.

Taking for W the point set of a homogeneous Poisson process PA ofintensity A, and letting A be the disc of radius r centred at the origin,we see that G A (PA ) = G, ,y More genet ally, in G A (1r) both A and 11/

211 Continuum percalat iaaa

may be chosen to lm random: one of t he simplest cases is when we assign

independent identicall y dish United non-negative random variables r(x):to the points .c of a homogeneous Poisson process 'P A , and take the unit

of the discs 13„,x)(3.), E PAClearly. all the models above extend trivially to higher dimensions

There me many other 'natur al ways to define random geometric graphs,

some of which we shall mention' in Section' 2If we condition on a particular point c: E 112 being in PA , then the

degree of c in G, ,A has a Poisson distribution with mean riT 2 A lit rad

the sir tutu e of G, ,A depends on the parameters 1 and A only through

the expected degree: for Aurg A t 11): , the graphs A„ and C„ A, have

the same dish ibution as abstract ,midair' graphs Thus, when studyingfin exannile, the various critical phenonwna concerning these giaphs,

are free r1) change either / on A, provided we keep a = / 2 A COnSrallr Inview of t his, ice shall \\ /ite (7(a) fu, any of the random graphs C, A with::

o = -at 2 A, the canonical neptesentative being C, we call a the degree

of 0, A The quantih a is also known as the connection arca. OF ShIlply,

area: G, is ((ethicl bi joining each point x E P, to all othei points of

P, in a disc with area

hi an obvious sense. the random graph CI, A models an infinite corn

mimic:at:ion ! Rawer k in which two transceivers can cornmunicate if their',

distance is at most 1

As in the discrete case (when studying percolation On lattices or

hit (ice-like infinite graphs), we say that (.7, A percolates if it has an in-

finite component N'Ve unite 0(t A) = 0(a) for the probability that the

component of t he or igin is infinite To intake sense of this definition_ we

shall condition C, ,\ on the or igin being one of the points; equivalently,

we shall assume that the origin is in 'P A This assumption does not

change the distribution of the remaining points of C, A

The first question concur Mug I his ; continuum percolation' is when the

'percolation mobabiliti s 0(t, A) = 0(a) is strictl y ppositive. It is trivial.

that 0(a) = 0 if a is sufficientl y small and 0(a) 1 as a Since

0(a) is monotone ha:teasing, there is a critical degree or ci Meal area a,:

if a < 0, then the probability t hat C7(a) maculates is 0, and if a > at„

then this probability is strictl y positive Also, as inn the discrete case.

Kohnogoloy 's 0-1 law (Theorem I of Chapter 2) implies that if a < 0„

then every component of G(a) is finite a s and if a > a, then C.;(0) hasan infinite component a s Note that I he critical degree at cot responds

to the critical probability For percolation As we shall see later in

Theorem 3, the natural analogue an of :/ ci is equal to a,.

S I The Gitlie,t disc model 245

Our main aim in the test of this section is to give hounds on the n it-ica t degree a c . An easy way of bounding a, is by e0111pat ing C, A to adiscrete percolation model with good bounds on its critical probabilityPerhaps the most natural model in the face immolation On the hexagonallattice in which each face is open with the Sallie probability indeipen-(entry of the states of the other faces, and closed odic' wise; two faces ateiwighbouts if they share au edge (As every vertex has degree 3, some-

shat misleadingly, this is equivalent to sliming a vertex ) This amdel

is t ome usually desetibed as site petcolation on the triangular latticeTheorem 8 of Chapter 5 tells us that if p < 1/2. then a s there is noface petcolation, and if p > 1/2, then a s we have face percolation

To obtain a face percolation model F l om PA. let A be a hexagonallice, with each face a regular hexagon of side-length s. Define a

face percolation model on A b y setting a face to be open if it containsi t least one point of PA. and closed ()diet wise. so each face is closedwith limitabilit y c -AA and open with probability t - C AA , \viten, A =3452 /2 is the area of a hexagonal face (see Figure 3) Front Theorem 8

010Figure 3 Comparison between Gilberts model and face percolation on thehexagonal lattice, i e site pwcolation on the hi:log[11w lattice: a hexagon isshaded if it contains one or more points of the Poisson process The co p e-sponding site in the triangular lattice is then taken to be open If the hexagonshave side-length s. then AB = 2 ‘,/s. so AC =

of Chaplet 5, if 1 - c -A1 < 1/2 then we do tot have face pet col ion,and if 1 - c -AA > 1/2 then we do

All that remains is to thaw the appropriate conclusions alma the

disc model G, A Note that any two points in neighbouring faces ate

at distance at most ((2A 212 ) i/2 s s n/171, and any two paints

iu non-neighbouting faces are at distance at least H Consequently, if

246 Continuum percolation

1 — e- A•1 < 1/2 and r < s then we do not have percolation in C,,A,and if 1 — c- AA > 1/2 and r > sVT.3 then we do, Taking A = 1 andsubstituting A = (3/:73/2) 52 , we find that if (3 n41/2)s 2 > log 2 thena„ > 7r5 2 , and if (342)52 < log' 2 then ac < 137,s2 , i c ,

log 2 c, -Mil-1o°' 9 < a, <34 34

We have given these inequalities to show what one can read out of theapproximation of a Poisson process by face percolation on the hexagonallattice, although the lower bound above is worse than the trivial boon1 implied by the simplest branching process argument. On the otherhand. the t ipper bound, given by Gilbert [1961] under the assumption(unproved at the time, but proved now) that the critical probability forsite per colation on the triangular lattice was 1/2, has not been improvedmuch in over four decades The slightly better upper bound in thetheorem below is due to Hall [1985b], while the lower bound is the boundfrom the seminal paper of Gilbert [1961]. (In fact, the numerical valuefor the formula given there was 1 75. !)

Theorem 1. Let a„ be the critical degree (area) Pi the (tither (Boolean

,lice model G,. Then

671

271-

Proof Let us start with the lower bound. Let Co he the vet tex set ofthe component of the origin in G, = G, , r, briefly, the component of the::or igin We shall use a very simple process to find the points in Co oneby one, but in order to achieve a concise formulation of this pr ocess, wedescribe it in a rasher formal way.

We shall constr uct, a sequence of pairs of disjoint (finite) sets of pointsof the plane, (Do, Lo),(D 1, say The points in D, are the pointsof the component Co that ale dead at time t: they belong to the com-ponent Co, and so do all their neighbours; the points in L 1 are hue attime t: they belong to Co, but we have made no attempt to find theirneighbours. To star t t he sequence, we set Do = 0 and Lo = {X0 }, where

X0 = 0 is the origin Next, let No be the set of neighbours of Ko, andset Dr = {X0 } and L i = No Having found (D,, L i ), if L, = 0 thenwe terminate the sequence; otherwise, we pick a point K, horn L I , anddefine Dt+. = Dt U{Xt } = {X0 ., X, } and L 1+1 = u L,\ {NJ,where Nt is tire set of neighbours of Kt that are not neighbours of any of

612 < (lc < 10 588

8 1 The Gi(bell (list model

247

the points in D, Since is locally finite (no disc of radius l' containsinfinitel y many points of P i ), the sets D, and L, al e disjoint finite sets

By construction. U C fun then mole, if =0 then Ca =

Since

\ I.V0 1 At C U Ari=0

laVe

ID / I = t — 1 <

(1)

Let 1 7, be the disc of /minis I with centre X t , and set U, = (1,0

Conditioning on the points Xo. .X1 . we find that IN / l is a Poisson,

andom vat iable with wean II ./ \U,_ 1 1 Since the centre K t of I", is in, a

disc V,. s < t — I. Figure -I tells us that

Figure 4 Two (lists V, and I; of radius r with effi g ies A. and Xf, such that.E The (nett of Pr \ V1 (shaded) is maximized if dist (X„. X t ) ttct I, as

shown In this ease the area is ft = (if + 4-1+t 2

Now. to bound let Zo. Zi. be independent Poisson random

variables. with E(2 0 ) = a7 2 , the men of the disc Dr,, and E(Zi ) = b furi> 1 Then inequality (1) implies that

ik-POCol > < P

\i=0

II 6 < it follows easif\ that 1 ICil l � as A i . )

9-18 Cortiirtir II III Ile IrOla 1011

Since r

= 71 - < + = 2if + 3 \ /73

we have b < the ni tnat area is indeed at least as huge as claimed

Pot time upper bound on a u , Flail (19851,1 tweaked Ciilbe i Ls argumentr km2that gave die trivial hound 't'3,111 — 19 89 by replacing the cells of a

hexagonal tessellation be 'rounded hexagons' To give the details of thisargument, consider the lattice of hexagons, each with side-length I Foreach hexagon H, let H' be the rounded hexagon in 11, the int ersectionof the six discs with centres at t lie mid-points of the sides and touchingthe opposite sides. 05 iu Figure 5 By coast/ uct ion, each of these discs

Pigmy 5. The shaded part of each hexagon is the legion within distance I of

the midpoints of all ti sides The labels correspond to those in Figure

has marlins 15. so am two points of two neighbouring rounded hexagonsale at distance at most 2 N/if Hence. if the probability that a 'minded

hexagon contains at least one point of the Poisson process PA is greaterthan 1j2 then. appealing to Themeni 8 of Chapter 5 as berme, we findthat ary/ii.A Petcolates with strictly positive plot/ability therefore, ifthe area of a rounded hexagon is a then C .—A" < 1/2 implies that a, <1.27A Hence.

97 log 2<

Finall y. the men a 01 a rounded hexagon H' can be mead out of

Figure 0 First the area of the sector ACD is 3c/2 To calculatelip . note that sinyi, = r.}ir; sin(57/(i) = so cos; = n/15/4 Also.

S / The Glebe') disc inodd

249

Figure The points A and C' are midpoints of two opposite sides of a hexagonII; 0 is the centre of H and 131' is half of a side. Thus, A0 = 01' = V3/2.013 = BC = 1/2 and LCOB = 7/6 The shaded region is part of a circlewit It centre A, so AD = We write y for LIDO and r5 for sothat ,42, + = 7/6. The shaded domain DOC' is one twelfth or the area of arounded hexagon II'

sin 12 sin — ) = --)f)-'45T \ir") --- I). while t1'= II esio( ) = 0 2709Fin t lumina e.

sin cOD = AD — (1-7)

sin(57/6)

and so the area of the triangle .40D is

(Bc,OD) = 31.-'3(15). —1) OD /I

The mea a of ' is twelve times the area of DOC'. so

13\41.(Ji

:32— 3r--1 tncsin( )

91:3( \/-7) —1

8

so that u = 2 -167 and a r. < ID 588 . as claimed.

The simplest form of the branching process a/glutton' above implies

that if .1 is any bounded open svnunettic set in 1P1( 1 . then the criticaldegree fin C T (PA ) is at least i

IGdI [1985b] improved s lower bound as well: this inunoventent

involves a more substantial modification of Cilbez argument than the

tweaking of the upper bound p l esented above. Indeed to obtain his

improvement.. Flail compares disc percolation ' to a multi t ype 'munchingocess, with the 't y pe' of a child defined as its distance hour the father.

Them em 2. The radical degree for the Gilbeil (Boolean) iliac model is()realer than 2.181.

259 ermithumn percolation

There has been much numerical work on the critical degree For theBoolean model, and on its square yin iant Simulation methods have been

used For the disc by Roberts [1964 Don't) [1972], Pike and. Seager [1971],

Seager and Pike [1.971], Flenrlin [1976], Haan and Zwanzig [1977], GawL:inski and Stanley [1981], Rosso [1989], Lorenz, Otgzali and Heuer [1993],Quinlan:ilia. and Torquato [19991, and Quintanilla, Torquato and Ziff[2000], among others, and fort the square by Dubson and Garland [1985];Aron, Drmy and Balbog [1990], Garboczi, Thorpe, DeVries and Dm.,

[1994 and Baker, Paul, Sreenivasan and Stanley [2002] (Not suupr is

night. , several of the bounds obtained happen to contradict each other )For the critical degree of disc percolation, Quintanilla, Torquato andZiff [20001 gave lower and upper bounds of 1 51218 and 4.51228; forsquare percolation, Baker, Paul, Sreenivasan and Stanley [2002] sug-

gested 1 388 and 4..396

In studying the critical degrees in these two models, Baliste" , Bollobrisand Walters [ 2 005] gave rigorous reductions of the problems to compli-cated mune] ice! integrals, 1\ hich they calculated by Monte Cat lo meth-ods Indeed, their method was the basis of the discussion of rigorous 99%confidence intervals For site and bowl percolation critical probabilities in

Chapter 6 The basic idea is to use results about k-independent per cola-Hon to prove that a certain bound On holds, as long as a certain eventE defined in ter ins of the restriction of 0, to a finite l egion has at leastsome probability Thi (For example, they consider the event E that thelargest components of the subgraphs of (7, induced by the squares [0, ((2and [1 . 2/1 x [0. L] are part of a single component: in the subgraph inducedby 10,2(1x (0, t] Considering events isomorphic to E, there is a naturalway to define a Lindependent bond percolation measure on Z2 such thatever V bond is open with probability P(E). Another possible choice haE is described in Chapter 6 ) Unica tunately, one cannot evaluate P(E)exactly: however, there are Monte Car lo methods for evaluating P(E)that conic with rigorous bounds on the probability of errors of certainmagnitudes; see Chapter 6 for more details of the basic method Usingthis technique. Balister, Ballobris and Walters [2005] proved that. withconfidence 99 99%, the cr itical degree for the disc percolation is between4.508 and 4 515 and that fin the square percolation' is between 4.392and 4 398

The results and methods we discussed in ear lier chapters for discretepercolation models are easil y applied to the Gilbert Boolean disc modelto 'rime the Ind( 11/01 letiti of the infinite open cluster above the critical

8 I The Gilbw t disc Model

degree a c. and exponential decay below a, We start with a a result ofRoy [1990] giving exponential decay below a„. which implies the analogueof pH pr for Gilbert's model This result holds in any dimension; lotnotational simplicity we state and prove it only for dimension two

Theorem 3. Let 7c. f 2 A = a < ac . where a„ is the critical degree for

the Gilhevl model, and let ro (ar A H denote the Immix, of points in the

component of the origin in G, A Then

P0C0 (G, A )] > <

where c„ > 0 does not depend on n In pal ticular a L. where

ur= thr {71/ 2 A : IrtaCo(G, A lp = Do}

Proof 'The result is more or less immediate from klenshikov's Theotent(see Theorems 7 and 9 of Chapter 4), using the natural approximationof continuum percolation by discrete percolation Roughly speaking, weshift the points of the Poisson process PA slightly. by ;rounding theircoordinates to multiples of a small constant S If the shifted pointsare at distance at most r - then the original points must havebeen connected in C, A On the other hand, if the shifted points are atdistance more than r + 20.6. then the original points cannot have beenconnected in C, A

Let us fill in the details of the argument. Pick A i > A and rr >so that a < < o r , and 6 > 0 such that r + 2015 < r I Let6722 = (62.7) 2 lie the square lattice with laces

Fit = {(,c, y): IS < < (i < y <(J+ 1)6}.

Let A(6,./ j ) be the graph whose veil are the faces Flci , in winch twofaces are joined if the maximum distance between two points of theirunion is at most r i We couple G,, A, with independent site percolationon A(S. L I ) in the obvious way: declare a site F1 5:, of AO, to be openif and only if it contains at least one point of 'P A , Note that the statesof the sites (i e laces) are independent, and that each is open withprobability pr = 1 - c-Ala

Suppose that there is an infinite open cluster in A(6,1 j ). If :r and yate adjacent open sites of AR I I ). then there is at least one point of PA,in the faces of ,5& corresponding to A and g. and any two such pointsare joined in CI, , ,A , . Hence. G,, A, has an unbounded open cluster Butor < a„. so this event has probability zeta Thus, p i < pli(A.(6.11))..

2519_5

9'0

Continuum pettolalion

The gul l) ) ) A O I I I satisfies the conditions of Menshikov's Theorem,

Theorem i of Chapter (As usual we tepid A(6.1 1 ) as an oriented

graph In replacing each edge by two oppositel y oriented edges ) Set

p = 1 - e- A<C2 p i If each site of A(6, 1 i ) is now taken to be open

\lit II wok/tinily p, independentl y of the other sites, then by Theotems 7

rind 9 of Chapter 1. we have exponential deca y of the tartlet of sites in

the open cluster of the (night in A(S. r r)

net uming to Gilberts model, this time we take a lace of 6:2 to beopen if it contains at least one point of PA. so the faces me open hide-

pendently with probabilit y p Vie condition on the (night l y ing in PA

If two points ()I PA ate \\ ithin distance I then am two points of the

couespondin faces ale within distance + 2 6 < !I so the fates are

joined iu AO./ Thus lot every vertex of C r A in the component Coof the might, the co/responding face is in C(,, the open clustet of the

(night in AO, II) Given the unwire/ of points of PA in each lace of

el, has a Poisson distribution conditioned on being at least I An easycalculation shows that the total number of points in such laces is very

unlikely to be much beget than IC11. and the result follows

irleester and R i o i19911 purred uniqueness lot both the occupied and

vacant clusters. Again, this result holds in any dill/C11:60/1

Theorem 4. In Ihr Gilbert model G, A n s There iv at most one

compor t ( rd. and ol row,/ WIC 11111101111(led component of E, A q

Unlike Theorems 3. this does not follow easilr limn the co/les/Raiding

discrete results: the Brent that them is at most one unbounded (occupied

01 vacant) cluster is 'within increasing not decreasing . there is no

stotight fo t w ind wa\ to bound its probabilit y b \ that ( )I an event iu a

disown. , app t oximation to ()illicit 's model However. the Button Keane

proof of the Aizenman ICesten Newman uniqueness result lot lattices

(Themen t 1 of Chilling 5) can easilr be adapted to Gilbett's model

(hice we know d int (with probability I) G, A has a unique infinite

component-. it is reasonable to expect that inside a -big box' them is

onl y one -big component. Pen t ose and Pisztoi a [1990] showed drat the

heuristics. if moped r stated, ate indeed tine. but they hare to work

rather Laud to more 1110111

Ot Ile/ results tot percolation on lattices have analogues lot cumin-

mn pe t ( M i tt km Fro example, Alexander 0996] proved an rutalogne

(111? Russo SCVIIRMI SVelsh liniment for occupied eh/stets in G, A.

8I 7Ite Gillett disc model 253

e 101 13, A N. C( tesponding result lot vacant clusters was plover( by

Boy [1990]

d extelisionsEven in the plane. t he 011)00 model has 1111111(1 otts tur

Em example. instead

define a random set

if a dim... We way use a convex domain IC C R2 to

ti

D a . = U(

IC )i=

whew PA } is a Poisson process of intensity A in the planeThis process percolates if D k. A has an unbounded component Note thatK is not assumed to be s we genet alize the I/11(1 etat ion ofthe Gilbet t model shown in Figure 2 The co t tesponding giaph is exact IvG A (PA ) as defined eat lief wit h A =I( —k In pm ticulal. fhe expecteddegree of a vet tex is A times the m et' of IC — IC \ (IC rotthe critical intensity of this percolation process. lonasson [2001] proved

the beautiful result that, among convex (10111/1illti IC of area A, (K )is minimal fin a triangle Roy and Tairenuna [20021 showed that ananalogous result holds in higher dimensions, with a simplex replacingthe triangle

To conclude this section Nye note that thew turf natural models with

critical degrees close to the minimum First, as shown by Pent ose

the finical degree for long purge pe t colat kW " ill Z2 (ot J') tends to 1

as the 'range' tends to infinity (See I3ollobas and Kohayakawa [1995]

for a combinatorial pool.) 'Molesurprisingly. let A : the annuluswith twin 1 and I + e (anti so area rr(2s. E2 )); then the critical degreefor (Py ) tends to 1 as E 0. This was proved. independently byFlanceschetti, Booth. Cook. Aleeste/ and Duck [2005]. and I3alistet.Bollobtis and NValters (2001) In the lattei. Impel. it was shown that theof tesponding asset t ion for 'squaw annuli' is false: let S, be t he squa t e

annulus with the buret square having side-length 1, and t he outer 10 < E < 1. say 'Then the critical degree for Cs_ ('PA ) is at least c > 1.with c independent of E

Tinning to hall pc/potation in higher dimensions (correspondingthe Boolean disc model in dimension two), let (14(!`i) be the critical degree(volume) in Ea , so that = (4.2) Peru ose 1[996] showed that HY)

d (pc., Batiste/.. Bollobris and N Yakut s (20011 proved a general result

about models with ethical degrees tending to I t he minimum possible

value: t his implies tiiyiallv the results fin annuli and high-dimensional

balls


Ilathet than asking fin an unbounded component, one may ask Mrall el almost all of out space to be coveted by a collection of randomsets Usually, the random sets a t e not translates of the same set, butare chosen with a certain probabilit y distribution For example, we maytake independent identically distributed compact sets K I ,16, in Rdand take the random set

E U(,r1 NJ,in I

when! {.1 _ is a random sequence of points in 1P d Conditions

impl y ing that E is the entity space R I were given by Stovall. Kendalland Mucky [1987], Hall [1984 Meustet and Ro y [1994 and Mo[clumpyand Scher balm\ [2003]: At linn a, Roy and Sat hat [200-1] gave conditionsfor the sets to covet all but a bounded part of

8.2 Finite random geometric graphs

As random geometric graphs fiequently model 'real-life' networks, e g anetwork of transceivers dist/ Hutted in a bounded domain, it is mum al tostud y finite random geometric graphs.. with the quintessential examplebeing the restriction of the Boolean model to a finite set To define this,let be the restriction of a Poisson process P„ of intensity n to thesquaw [0, Fur r > O. let G, (1;,) be the random geometric graph in

hich two points of V„ ate joinecl if their distance is at most I It is easy

to check that C, (1 7,, ) is close to the model C.', (U„).. in which U„ consists

of a points chosen uniformly noun the unit square, and two points arejoined as before Clearly, scaling makes no dillerence to the model:multiplying I by I and taking the testi idiot/ of the Poisson process,P„ 712 to ID, ([ 2 we get a model isomorphic to GC (1/1„) Fat example,it is natural to replace the unit striate by [0, jir] 2 and then take thetesniction of the Poisson process of intensit y I to this squaw AV hatdoes matter is the relationship between the expected degree and theexpected number of points

In fact, to obtain a mathematically more elegant model, we shallchoose our points front the torus .11 obtained horn [0,11 2 by identifying0 with C. To he precise, let G, A) he the random graph whose vertexset is a Poisson process PA on ff7 with intensity A, in which two points of

'PA are joined if their distance is at r The expected number of yer-Rees of this random graph is n = and fort tC/2 . that a E PA,

t he expectation of the degtee of a in CI, (Iti, A) is d = n 2 A Once again,

8 2. Finite noulom geometric graphs 255

the scaling is irrelevant: a and d determine (3, (r71, A) so, with a slightabuse of notation. we may write (17„ d for this model. The main advan-tage of using the tor us, and so C,, e h is the homogeneity of the model:however, to all intents and purposes, for r1 = il(n) o(n) the modelsCri„ (tin) and (3, (U„) are interchangeable provided d =7112n.

The first question we should like to answer about the finite graphsC„ , a is the following For what values of the parameters n and d is therandom geometric graph C„ a likely to have isolated vertices? This wasanswered by Steele and Tierney [1989] Later, Penrose [1997; 19991 ex-tended this result by giving detailed information about the distributionof the minimal degree ti(G„ a) of C„ ( 1 It is fascinating that this result isthe exact analogue of the classical result of Lidos and R 6/ IVi [1961Id onrandom graphs (see also Bollohas [2001, Theorem 3 5 .1) Here we presentonly a weaken for w of Penrose's thecae'''.

Theorem 5. Let rl = d(n) = log p ± A; log log n o(n) where Ar is afixed non-negative integer. If o(n) then

Nri(G„ < — I

and tf 0( ) then

(8(G „ > lr ± —

Pivot :10 simplik the calculations. we sketch a proof h=0 Thus.we set e = log p + o(n) : and let A and t satish = A1.1= and d = 71-12A,

so that C„ 6', (14, A) We write X for the number of isolatedvertices in C u m, A) The probability that a fixed disc of radius 1 in ticontains no points of our Poisson process Pt is c'n-A It follmvs hornbasic properties of Poisson processes that

ilit t A u) = ne-' 1Ape tic-cr (2)

Indeed, dividing T? into (C/r) 2 small squares Si of side ar < 1/2, bylinearity of expectation, 1E(X 0 ) is (I/s)2 times the probability that Sicontains an isolated vertex of CI, OTT A) (No Si can contain two suchvertices ) As E 0. this probability is asymptotically

so

li (1ltt/s) 2 AE 2 C -AS2— r".

256 Coolinuum pc, colotion

imph ing (2)

II (1 — then (2) implies t hat

P(S(G„ = > 1) < u( 1)

Now, suppose that (1(11) To prove that in this case (7„

(7, (T./. A) is ye t v liken to have isolated vet tices. we need mutt het Oat her

trivial) step IVe knou that ni = Iti(X0 ) — Ix: It is easil y checked

that t he second moment Itt(„V,.;) of Xo is oh, = (1 + o( 1 ))/// i ; hence. by

bi ltebvelly% l ts inequality

—INO(G„ a) > = Pau = u( 1

The genet al case can he p i (‘ ed along the same lines. \\ it 11 a little mote

calculation.

Im fact. it is easy to pttne that- Fin d = ((Oil in the pp/ Op into lunge,

thenuttbet _Ar. of Net t ices of degtee A. has as y mptot ically Poisson distri-

bution: t his implies Yen good bounds on t he mobability Il il ei(G„ a) >

Although this is not inintediatek obvious, it is not hind to show that

hemetn 5 does nuked cal y ore/ to t l i e model (3, defined on the

souffle l athe/ than the lotus, i e that the dumndan effects do not

matte]

ou t nex t aim is to state a considembh weight jet lesnit of Pentose

establishing a close connection between t he mope ' ties Of •4-runitecteduess

and of l i ming minimal degree at least s e star t br defining t he hit ling

radius lot at at hitt tu n ptopett y and an al bin set Chen a point set

P and a monotone inneasing propel Iv Q of graphs lie a I n Opelty Q

such that if (7 has Q and C' is obtained In adding an edge to G then

C' also has (9) the !Oiling !who .: pQ (P) of (9 on Pis defined as

pQ (P) = whiff : C,(P) E Q}

Note t hat if Q C (hell P(2( > 1 )(2'(P) f or PVC? ty set. P In pm t ic-

ulat . the hitting radius of s-connectedness is at least as huge my t hat of

htrc ing minimal degree at least s:

)

lot f PC I o P

A basic Jesuit it t lie them \ of randrun glitplis is the result of 13ollobtis

and 7itonmson fltS51 that lot almost even I //mIum graph process t he

hit ting time of s-connectedness equals the hitting title of having minimal

8 :2 Finite random a t:met t le 'myths 257

degree at least s (see Bollobas [2001]. Theorem 7 -1) Penrose [1999] (seealso Penrose (2003. pp 302- 3051) showed that this result car ies over to

the Boolean model on the tor us

Theorem 6. Let PA be a Poisson process on the ton If with inten-

sity A Then

lint (PA) = ps>,(17 A )) = 1 q

Thus, for A huge enough, if we strut with a set PA of isolated points,

and add edges one by one, always choosing the shortest possible edge

to add. thew with high probability. the very moment this graph hasminimal degree 5, it is also s-connected

Combining this result with Theorem 5 we can identif y the critical

degree for s-connectedness

Theorem 7. Let s he a freed non-negative i ntent ' , awl let d = (1(0=log + (s — ) log log f t ± of ) 0.(n)— then

IP'(G„ :I is s-conneeled) —4

and if (1(0 — then

d is ssivnneetcd) —

As shown by Pennose [1999]. all these results can's over to random

graphs on the snuffle and. nuttat i dis nuttotis to the / i f-dimensional cube

10, lf"' and torus The analogous problems For the simpler graph de-

lined using the If.„-distance, rather than t he Euclidean distance werestudied by Appel and Huss() [1997; 20021

The random graph (7„ d above is connected if and only it . the minimalweight spanning nee in the complete graph on the same vertex set.

wit h each edge weighted by the distance between its end-vertices. hasno edge longer than d. Such spanning trees, generated by a (possiblynon-uniform) Poisson process in the m-dimensional cube [0,1]"'.. have

been studied by a number of people, including Henze [1983], Steele and

Slump [1987]. and Kesten and Lee [1996]

Our next aim is to studs the connectedness of one of the many modelsof random geometric graphs related to t he Boolean model. Let bethe restriction ()I a Poisson process of intensity l to a square S„ of urean and join each point r E to the A. points nearest to it. Let 11„ t be

the random geometric graph obtained in this way Note that o is the,e,rpected number of vertices of H„ ,e; also, if H„ . p has or > 1 vet (icesthen it has at least km/2 and at most km edges

The random geometric graph H„ k is again a model of an ad hoc::network of transceivers and, as such, it has been studied by many people,:including Kleimock and Silvestet [1978], Silvester [1980], Flajek [1983]Takagi and kleimock [1984], Hon and Li [1986], Ni and Chandler [1994]Gonzales-Banjos and Qniroz [2003], and Xue and Kumar [20041.

We should like to know for which Functions k = k(n) the graphis likely to be connected as n . More precisely, we should likefind a function ko(n) such that, if F.- > 0 is constant, then

lim P(,A- ro

is connecte ) =if k 5 (1 — 04.0(4

if k (1 + E)ko(o)

Such a function ko(n) is considerably harder to deter mine than the ctiti-:cal deg/ ee d for connectedness in the Boolean model given by Theorem 7;since the trivial obstruction to connectedness, the existence of an iso-lated vet tex, is ruled out by the definition of H„

As we shall now See, simple back-of-an-envelope calculations give usthe order of /,.[) (o) Nevertheless, we ate very far from determining theasymptotic value of 1;0 (0 In the arguments that fellov,, the inequalitiesare claimed to hold only if o is sufficiently huge.

Let us see first that if r > e then k = k(n) = [clog is an upperbound fix ko(n). If every disc of area a = 7r1 2 centred at a point a: E V„contains at most k other points of S. then ti„ e contains „ as asubgraph, whew „ is the graph on V,, obtained by joining two pointsif they are within distance r The variant of Theorem 7 for the square(rather than the torus) tells us that the graph C„„ is connected whp(with high probability, i e., with probability tending to 1 as n x) ifa = log n + log log o, say. (In the application of the theorem, s = 1, andev(n), log logn ) Fm this a, the probability that a fixed disc of area a.contains at least A' + I points is at most

ah±1 1 < ([/(!)/ <(k + 1)!

where c < < c and z > 0 nom basic proper ties of Poisson processes,it Follows that the probability that the disc of area a about seine point of1 ,7„ contains more than A. other points of V„ is at most 1E' Consequently.H„ , p is connected hp, as claimed

8 2 Finite modont geometric graphs 259

Next, we show that if z. > 0, then k 1(1 — s)logn/81 — 1 is a lowerbound for WO To this end, define > 0 hi= + 1, and consider a

family 'D of three concentric discs D I , D3 and D5 contained in S„, where

D; has centre x and radius it: see Figure 7 We say that the family D is

Figure '1 A family D = {D i , D. Dr,} of three concentric discs, and two pos-sible discs D, touching D I For k 2. the fandiv D is had., so containsno edges from the vertices in D I to the test of the graph

bad for the graph 1, (or set V„) if the following three conditions hold:

(i) D I contains at least k + 1 points of 11„,

(ii) D3 \ D I contains no point of V„, and

(iii) for every c E \ at , the region D, fl (D 5 \ D3 ) contains at least

+ I points of 1 1„. where D, is the disc with centre c and radiusdist(x..i) — 1

Clearly, if some family D is bad lot H„ t , then the graph k is discon-

nected. A family is good if it is not badLet us show that the probability that a fixed famil y D is bad fin H„ k

is not too small First, the probability that D i contains at least k +

points is approximatel y 1/2; an\ at least 1/3 Next soin thin (ii) holds

with ptobalrilit

Finally. condition (iii)) holds if it holds tin the points c with dist(i)..r),

tar For such a point c, the area of D, n (Dr, \ Da) is en 2 for sow

> 2 1111/S. one call cover D5 \ D3 by a constant number C of regions

R., of area 27/ 2 so that any D, con)contaitrs sonic The probability that.

a given R, does not contain at least k+1 points oft„ is Y(1) Hence, the

probability that (iii) holds is 1-0(1). and in pm Perrin/ at least 1/2 lot y

large Since the events (i), (ii) and (iii) are independent, the probability

that a family D is bad tor [In k is at least fr-H-5/(i

The square .5„ contains at least

>)- //

)0 -11 101(( — x) log n)/(87v) 5 log n

disjoint squirts of side-length lot so it contains at least this Walk"

disjoint discs of radius St Therefore. the probability that even- familt

D is good For 11„ p is at roost

( +16) lug!,

ConsequentIt the plObabilli V that 11„ k is connected is at most cii

of I)

The mg uncut above can he consirkriabli simplified if all we want

is that connectedness happens around k = Sflog 0: there is no need

to use Penrose s Theorem. Firemen ' 7, which is quite a big gun lot

such a S/11)111 sparrow Be that as it mat, the tenni/1(s above tell us

that if c 1/S then /1„ r„g „ ) is disconnected whp. and if r then

H„ Li l „; ,, i is connected whir :Kite and Kumar (200-11 were the first to

publish bounds on li tt (n): they Firmed that 0 07 . i logy is a lower hound

and 5 1771logn is an tippet hound. with the upper hound firllowing from

Pemose's Theorem (In fact, the upper bound 3 8597 log y is implicit in

Cionzriles-Barrios and Quiroz [20031 )

Bollobris. Sailun and \Villiers 120051 considerabl y Unmoved

the constants 78 and r in the Pi\ ird bounds above: in pm t

they disproved the nal trial coujectuc t hat kay) is asymptotically log y

which is the analogue of Theorem 6 fin H„ k

Tlieolern 8. If < 3103 Own II„ , 1 „„„i I s ( I l yto li nerlfrl wh it. I 11(1 if

c> 1f log I 0 51:19 fin?, „ „ H kfitlfleCh (I why

S Punic irutdoin geometric gtretlIP 261

Pot I he rather involved proof and a nundail of related distills \VC iteletthe reader to the ot iginal impel

Instead of asking lot the connectedness of certain million' geotnet ticgraphs. \ye may ask ha the mound domain to he cour ted by the t an-dont sets we used to define the graph. These enticing() questions havea long history. and ate prominent in three boolts: Stu\ an. Nendall and

Matte [1987]. Hall [1988] and Meestet awl Roy 119961 Here we shall saya few avoids about some of the man y results

lust before the dawn of percolation. D ymetzEN 119561 laised a beim-

Hiltl question collect ning cove t ing a citric la /tuition) 11(5 Let 0 <

(0. <1 Drop airs of lengths 1 1 . lg. independent IN at iondont

onto a circle with perimeter ro t w hat sequences (I ) do mu at cs cowl

the entity circle whp? Independent IN of 1)NowtzltN St eutel 119671 wiseda sindlat question and proved a 101mula that has trtnmid out to by yet vuseful Shepp 119721 gave a delicate at gument to in t hat a necessm \

and sufficient condition is that

,a 2(

Flat to [1973] considered iandoin arcs of the same length (r. 0 < 0 <(Sin pi isingly. this problem had been considered br Stevens 119391swei alyears byline Eho t etzlt N posed Ids question. in a 101 1 111a1 on eugenics.) Tostate Flat to's distill. let m be a fixed nal inal nundaa and (hop the at cson the (Tide (01 pet inwtei I) al Iiin(loni one la one. stopping ;is soonas CNC( y point of the cl i ck' is co N end at least ni limes \I) /he A„ „, 101

the /Hunk , ' of tics used (Thus. A„ „, is the hit ting time lot bayingan m-Fold covet.) Flat to showed that

litre 1111(;V,0-6

1 ( log( I /d ) loglog(1/d) th id)) = (:2"7(

\\ hich is once again itiminiscent of the classical Eldirs Ren y' resultSiegel [1979] plowed results about the distribution of thy length of the!Income( ' pall of the chyle: late/. Hall [1985a] extended this Jesuit to

higher dimensions. The t esult s of Stevens wen' gene/ idized Ir y Siegel andHoist [1982]

The number, total length m id sizes of the gaps left by the coveting

arcs were studied by Hoist and Hasler 119841 and Iluillet [2(103]. amongWhets Inparticular,Huillet made use of Stentels identity to Noy('exact and as\ nip( ot ie results about these quantities

A dilated problem_ due to 1 11 ()ns i 119581 conceins choosing disjoint

9 1i9 C'oatinaryrn pal colal ion

shot t subintervals of an interval at random: this problem and variants

of it gave rise to much research (see. e g . Nev (19021, Dvoietzkv andRobbins [196-II, Solomon and \keine' . [1986]. mid Coffman, Flatto and.delenkovie. E2000D; these problems are also studied under the name ofrandom sequential adsorption models

Turning to random covets in higher dimensions, Machina [1988] stud-ied the threshold lot random caps to covet t he unit sphere 8 2 C le'*Suppose we put N spite/War caps of area rEtp(N) independently and attantrum on 8 2 , and that

p(N)Nlam —

log A"

Machina proved that if r < 1. then whp the sphere is not coveted com-pletel y, while if c > I then whp it is. Further results of 'Mitcham 11090;200-0 concert the intersection gr aphs of random arcs and random caps:these graphs are the analogues of the gr aphs on the tor us andsquare we studied ear her in this sect ion

In a differ ent vein, Aldous [1989] used Stein's method to prove sharpresults about covering a square by random small squares

Gene] alizing several em rim results. Janson ([986] used ingenious and

long arguments to prove that. under rather weak conditions and al-er iippoquiate inalization. the (random) numbei of random small

set s needed to cover a larger set t ends tot he ext Imile-value dist ill n i t ionexp(-c 11 ") as the measure of the small sets tends to 0

TO conclude this section, we shall sa l a few words about the random

convex hull problem vet ,mother old problem about random points in a.

convex domain Pick a points at 1 ile(10111 horn a convex domain K CIE2.

NV hat can one say about the (r andorn) number X„ of sides of the con-

vex hull H„ of these II points? Renvi and Sulanke [1963] proved that:

the distribution of X„ depends very strongly on the smoothness of the

boundary: if K has a smooth boundary then Et-k„) = 0(1 1/3 ), while if

K is a convex k-goll titan Er(X„) = (log (1)+ L(C ) wrier(' Cis .Eulet's constant and c(K) = o(k) depends on K and is maximal finregular polygons and their shine equivalents In a follow-up paper,. Renyiand Sulanke [190-1] studied the area and perMiele/ length of the convexhull 11„ These results of Reny' and Sulanke have spurted much research,including papers In- Cimeneboom [1988]. Cabo and GI oeneboom 099-0,Using [199 ‘0, Hume! [199-1: 1909a; 1999b]. Using and Bing-ham (19981, Brinker and lising 119981. and Finch and [-ureter (2004] For

8 Random Vih onot percolation 263

example. Gr Oen/MOM/I pl (Wed that, as II the normalized ran-

dom variable (X„ — 27rc i a l/3 )1(n 0' V27o)) tends to a standard normal

random wadable, inhere(37/2)"-1/3 r(5/3) 0 538 and co is

expressed in tams of complicated double integrals Finch and Hnetet

gave an explicit expression fin c„

8.3 Random Voronoi percolation

In this section we shall considet Net another nuns per colation p i 0-

Criss associated to a Poisson process in 1Pil l for d > 2 Up to now. out

Poisson process was used to define a random geometric graph. and the

question was whether this graph has nu infinite component or not.. thistime we go farther: we consider a graph defined by the Poisson process,

and then considet site percolation On this graphTo be precise, let 'P lie a Poisson process (of intensity 1. say ) in RI

Fot even: Poisson point: z (i e E P), the Open Voronoi colt U„ of

with tespect to P is the set of all points closet to ,r than to am : other

Poisson point The closure of U. is the closed rolonth cell of

simply the Voronoi cell of with respect to P Thus.

= (P) = E : dist (y. ,r) < clist(y,x 1 ) for: ail .riE

It is easily seen that. lot a Poisson process P. with mobability I each

cell lir = ll,.(P) is a if-dimensional convex poh tope with (Mitch main(4 — 1)-dimensional laces, and any two Votonoi cells ate either disjoint

or meet in a full (4 — 1)-dimensional face Also, for any k-dimensional

face of ljr , them are exactly d + 1 — k Voronoi cells Vit containing it

Fot convenience, we shall assume that these conditions always hold We

call V(P) = {Vir. E PI the Voronoi tessellation associated to P. The

random tessellation V V(P) is 0 lundata Voronoi tessellation of Rd;

see Figure 8.The Voronoi tessellation defines a graph Gp on P: two VDT onoi cells

are ailjaceni, if they share a (4 — 1)-dimensional face, and we join two

points X, y E P if their Voronoi cells ate adjacent; see Figure 8.The terminology is in honour of Voronoi [1908] who, at the beginning'

of the last centur y, used these cells and tessellations to study quadratic

forms In fact, concerting tessellations in two and thee dimensions,

Dirichlet [1850] had anticipated \ To :tonal by ovet fifty years, so one may

also talk of Dirichlet domains and Ph Oil& tessellations In 1911, these

tessellations were again rediscovered (see 'Thiessen and Alter [19111), so

in some circles they go under the mane of Thiessen polygonalizations

9 64 Continuum p(vcakifion

l'ignie $ [lie upper figure slums pact of I he Voronoi tessellation I (P) asso-ciated to it P0i55011 process P c E* 2 : the points ()I P me also short n If theVoronoi cells associat ed inert t hen their common edge is pall ofI he per pendiculat bisector of ry -the lower figures slum t he graph Ccdal ed to I . (2): I 0 points ()I P arc joined if the cur responding Voronoi (ellsmeet On the left each hund of Cr is (Hoc ir rising two st might line segmentson the right using one I he first ewbedding (4 Gp is r lent lv plaua i II is east/to shore 1 hit/ t he second is also

S :1 Random Vowing percolai nn 255

Other selbexplanatm y terms are Voronoi dingrmn and Mt idled diagram

for the tessellation. Voronoi polygon and Di; ichlci polygon for a cell,

and Poisson- Voronoi lesscllation tot our raudour tessellation. In what

follows, we shall use the terms Voronoi tessellation and random Voronoi

tessellation

The stink of Voronoi (Dirichlet) tessellations has a vet v long histolv.

especiall y in discrete geometr V ill connection with sphere paekings and

other problems: See for example. the books by Fejes lath [1953; 1972].

Rogers (19611 and BOrOczky [200-0 Although detruministic problems

rue outside t he scope of t his book let us remark that I he sphere packing

pr oblem asks for the maximum density of a packing ol congruent spheres

in Ed In 1929 Bliclihrldt gave t he impel hound 2-(1/2 (d 2) for this

density Using all ruguinent suggested by H E Daniels. Ilogets [1958]

impuued this bound to a certain constant ad. With (Id 2-`'f2 d/c as

— x Twent y Yeats later_ Kabatjanskir and LevenStein [1978] gave

a bound which is better lot > 13: recently Bezdek [2002] proved a

lower bound for the sulfate area of Voronoi cells which enabled hint to

improve Rogers's bound for all d > 8

There has been much research on random Vol ° tu g tessellat ions as well.

first in civstallogiaph y and then in mathematics. For solids composed

of diner ent kinds (A cr vstals. Delesse [I8-18] estimated t he fraction of the

1, (flume occupied by the it Ir CI Nstal: a cent in N Into. Chaves [1 956] gm()

statistics fin these estimates. Johnson and Meld [1939] gme a model

for crystal growth: the cells of this model depend not onl N on a Poisson

process. but also OH tire 'arrival times' of the nuclei These cells need

not even be convex, although the \ ale star-domains lion their !nuclei

Meijering [195:3] int t oduced Voronoi tessellations into crystallography,

without being aware ol the considel ably earlier papers of Dirichlet and

Voronoi Since then, this model (attributed to Meijering. rather than

Dit bidet or \known) and the related Johnson Mehl !node] have been

much studied

Gilbert [1962] ' moved results about the expectations of the surface

area the nunther of faces. the total edge length. and other parameters

of a cell (poi Itedion) in a landon) \Ammi tessellation In pm titular,

he noted in passing that Enlels Formula implies that in the plane the

expected numbei ol vertices (r1 a pol ygon is ti

Since the 1960s. a considerable both of results has been proved about

the 'typical cell of a random Voronoi tessellation: see Moller R99-11

and Stu \ an Kendall and Med:v[1995] ha man y results. Here, we shall

266 Crontinuuni percolation

note only some of the MOHe recent results concerning planar tessella-tions Haven and Chine (2000) gave a formula for the probability thatthe cell containing the migM is a triangle, and Calka [2003] gave anexplicit formula for the distribution of the number of sides Calka andSchreiber [2005) proved results about the asymptotic number of ver-tices and the area of a cell conditioned to contain a disc of radius r as

Continuing the work of Foss and Zuyev [19961, Calka [2002a;2002b], made use of the result of Stevens [1939) mentioned in the previ-ous section to study the radius of the smallest circle containing the cellof the origin, and tire radius of the largest circle in the cell

Somewhat surprisingl y although Gilbert introduced Iris disc percola-tion 'nuclei, and studied the random Volonui tessellation as a model ofCrystal growth, he did not pose the problem of face percolation on ananchnn Vilronoi tessellation. However, a little later, in one of the earlypapers devoted to percolation (hear y, Frisch and Hammersley [1963]called for attempts to pioneer 'branches of mathematics that might he

called stochastic geometry or statistical topology'. Eventually, this chal-lenge was taken up Inv physicists; for example, Pike and Seager [1971]and Seeger and Pike [1974] per fornwd compute" analyses of the percola-tion and the conductance of the Gilbert model and some of its variantsDetailed computer ' studies of these models were carried out by manypeople, including Hall!) and Zwanzig [1977], Vicsek and !Kw tesz [19811,Gawliniski and Stanley [1981], and Gawlinski and Redner [19831 amongothers

Percolation on random Vonmoi tessellations was studied by Hatfield[197$], Winter Feld, Scrken and Davis [1981), .1cmuld, Hatfield, Scr ivenand Davis [1.9841, and kraut* Striven and Davis [1984]. In panticu-Ian, based on compute ' simulations and the cluster moment. method ofDean [19631, %\ i interfeld Striven and Davis estimated that 0 500 ±0 010is the critical probability for random VO/Olkai percolation in the plane(to be defined below) Nevertheless, no proof was offered even for thesimple fact that the critical probability is strictly between' 0 and I Aswe shall see later. this is not entirely trivial, unlike for lattices such asthe example S in Chapter I

In a series of papers, Valddi-As1 and Wier man [1990; 1992; 1993] stud-ied first-passage percolation on random Vononoi tessellations More re-cently. Gurvnel and Gliffitath [1997] proved substantial results aboutrandom \km onoi tessellations: in the problem thew consider. the lacesare coknued at /widow with I colours, and then the colours change ac-

S.3 Random. Voronoi percolation 267

cording to the deterministic, discrete-time rules of a cellular automaton

The question is what happens in the long run

Here, we consider cell or face percolation on the random Voronoi tes-

sellation Vassociated to a Poisson process P on le of intensity 1: eachd-ditnensional cell, or face, V. x e P. of V is open with probabilityp, independently of the other faces See Figure 9 for part of a random

Figure 9 Random Voronoi percolation in the plane at p = 1/2

Voronoi tessellation of the plane; the 01)(1/ faces are shaded Ecpriv-

alently, we shall study site percolation On the random graph Op, in

which two vertices ;r, q E P we adjacent if their Voronoi cells and

meet in a (d — 1)-dimensional face In this viewpoint, given P, eachsite of Gp is open with probability p, independently of the other sites

268 Co PI ipuu pt pet (plat lop

Since P has (a s ) uo ticc mu t ilation points, the graph Op is a s locally

finite, infinite m u ll connected

This set ting is rather different front that of previous chapters: we are

now considering percolation in a random NI vi ton went. i e.. stud ying a

random subgraph of a graph that is it sell tandom The fundamental

concepts of percolation on fixed paints cat ty me t vet N . natm all to this

set ting: we shall go met the basics in detail.

Let us define a little mot e formall y the probability measure we shall

use Let .1), Px be a Poisson process on 1P1 with intensit \ \ (Usually,

and without loss of genet alitv, we take A = 1 ) We shall assign a state,

open of clo‘ced, to each point of P so that, conditioning on P. the states

of the points me independent. and each .1' E P is open with probability

p \Viitiog P + and P- lot the sets of open and closed points in P,respect kel t% lion a fin nad definition it is simpler to (fist define P .' mid

and then set P to be then union

To spell this out. let P -1 and P- be independent Poisson processes

on -Rd w i t h intensities p\ and (I — respectively Prow the basicproperties of Poisson processes, or from the (roust uct ion given eat Her . it,

is emu 10 check that and P + ate disjoint a s tun Cher mot e.. given P

U P each point of P is in with probability p. independentl y of

the other points in P w rite ho the probabilit y measure

associated to the pith (P +). and Pp ha Pr p : to avoid clutter we

suppress the dependence on the dimension d. which will always be cleat

hour the context. The ptolmbilik measure is defined on a state

space fl consisting of oarfraturdions = (A - I . ). whet e X + and A-

are disjoint discrete (i e without accumulation points) subsets of

The J7-field of measurable events is inherited how the constr net ion of

the Poisson process \\Then talking of P and Op. we shall often ignore

events (It plobabilih 0: ha example we say that Op is infinite rather

I han infinite as.

Matt\ of the basic concepts of per colation ttanslate naturalk to the

andom Vo l carol context. although in some cases there are two descrip-

tions one gt aph-t hear etic and one geometric. Let us colour a \aaonoi

cell black il the point c E P is open. i e if E P . and while other-

wise \ Ve say that r E is black if a' lies in a black cell. i e . it c E F-

tot sonre z E E Rd is whitc if it lies in a white cell: see

Figure 9 'Chas, a. point .r E Pl a is black (white) if and ooh if a point of

P at minimal distance flows is open (closed) Note that points I hat lie

in the laces of the Volonoi cells ma 't be both black and white Ic w hat

8.`f Random inonoi percolation 269

follows tic shall mostl y write IC lin a point of P. awl o for apoint of ifi'l

As usual an open cherlci is a maximal connected subgraph of the

graph C-p all of whose sites are Open Alternatively a black elastic ! is a

maximal connected set of black points in Rd i e . a component of the set

.rt E : black} ()In twstin i ptions on P ensure that two Votonoi

cells l', arid meet if and only if they share a (d— 1)-dimensionallace.

i e if tool onb if :de E E(C7 Hence. open clusters and black clusters

are in one-to-one t espondetwe

Let Ei„ he the (tient that Op contains an infinite open duster Note

that PALL,: ) is au increasing lunction of p Also. slime P lots ne ac-

cumulation points. E r., is exact Ir the dent that t here is an unbounded

Itlack dust 01

FO/ ant L. the event IL, is independent of the values of P t"n [—Land P-11— L. LP' Hence as PI' is the union of tlw independent Landon)

sets P ±+ 1 ). 0,1 E the event E„,_\ be viewed

as a tail c/,0111 in a product limitabilit y space Kohnogwor s

04 law (Hutment I of Chapter 2). PI 1,(E.,) is () or I fin an y p SincePi,(E.,) is inciensing. it foliar\ s that there is a fin = E such

that

if p <

Iilp> pn

Since Op is locall finite, infinite. and connected as noted in Outp-

ut] I. there is a critical probabiliti ////(Gp) = pi i (Cp) associated to

site percolation on (3 p Note that ',OC R ) is a buolow variable, as

it depends on the Poisson process P How (3) if p < mi have

Pe (E, Hp) = 0 a s . and. for p > ps i we lime C'p) = 1 a s

Thus

Pn(C'p) = Pit 0 s

i e the I nidcun satiable pil(Gp) is essentially deterministic We sh a llcall p H the ci !tient probability fro random i'wonoi putrid& ion innoting that p it is also the critical plobabiliti rot site percolation on

almost //Vet \ VOInnui tessellation associated to a Poisson process in P1't\\c next tin n to the percolation purbabiliti ll(p) l\ it h probtrhilicr 1.

Use or igin lies in a unique Voionoi cell io E P. If co is open, let

Cf; he tire open cluster in Op containing the site to Also let Co be

the Had, diode/ of the might i e the component of black points in Trid

containing the or igin II Co is closed then both awl Co are taken to

(3)

270 Continuum pc [rotation

be empt y Note that Co is the union of the cells I E eff Set

0(p) = IPp (cii

) 1P1, (C is bounded)

Since 0(G p) = 0 101 p 0 for p > pu(ap),flt on i (T) we have 0(p) = 0 if p < pit and 0(p) > 0 if p > A i , just as forsite or bond percolation on a fixed graph

Let us note that, unlike pu(ap), lot a fixed p > [ (1) the percolationporbability p) associated to ap is not a constant but a tandomvariable Let us consider d = 2 fin simplicity; we shall see late/ thaty 11 (2) = 1/2. so. Cot p > 1/2, we have O(n) = E(O(Cdp i p)) > 0 It is easyto see that, fin any 1 I bete is a positive probability that ap contains kdisjoint triangles 1), each of which surrounds the origin Whenthis happens. we 11.M . C1 Nap: p) < ([ — (1— p) 3 ) 1'. : if all Once sites in any

are closed, then there is a closed cycle stalemating the et ight, and Cois bounded It follows that, even fin p close to 1, the /widow vaiiahle0(ap:p) takes values Arbit/tu ily close to 0 with positive pr obability Inpa l ticula b 0(C my; p) 0(p) with positive probability, so 0((dp; I)) is notalmost surely constant In the other ditect ion. it may happen that thereis one (11101 /110115 \colonel cell centred on the origin. with hundreds of

neighbows mound it: in this (1118e, lot any fixed p > 1/2. the !widowvat intik O(Gp:p) can presumably take values al bit/ to Hy close to

As we saw i l l Chaplet I, for percolation on lattices. it is I t kin! thatthe ethical probabilit y is bounded au a \ bout 0 and I As we teinalIced(nutlet , the cot tesponding tesult fin 'widow Volonoi pet colatien is notemit el% trivial The approach that [list sin lugs to mind is comparisonwith independent percolat ion 01/ a lattice: rye applied this to Gilhett'smodel in the previous section Unlit' Innatel y. the tessellation 1 - (P) canand will contain at bitIwily large cells Thus. given any two points .r. y E

Fr t . there is a positive probabilit y that .r and y lie in the saute Voionoicell so the events • r is black' and 'y is black are not independent Onecan get mound this problem by continuing talld0/11 \TO/ onoi percolationto a suitably chosen 1-independent percolation model, and consideringthe event that the lest/jet ion of (P i-H.P-1 to a certain local legion I?knees eel rain points in Eild to be black. whatever P+ and look likeout side I?

Lemma 9. ho, mum d > 2, We ha vc p ii (d) < I

S 3 Random l'oronoi petrolation 271.

Pion{. \\le shall l escale, considering a Poisson process P = PA oil IRdwith density A much huger than 1

For each bond c of the graph Ed , let L 1 , be the corresponding line

segment of length 1 in Ra , and write

R,, [4 112) E : ELF. dist (x, q) < 1/2}

lot the (closed) 1/2-neighbour hood of L.Fixing the bond e fin the moment., we call covet L. by three closed

balls Bcf , i = 1,2,3, of radius 1/1 whose interiors ate disjoint; see

Figure 10 Let 13,, be the corresponding balls with the same centres

but radius 1/2

Elgin ° ID Three (closed) balls i = 1.2 ,3. of ' adios 1/-1 covering theunit line segment and the corresponding halls C H i . with

1/2 (labelled only foi i II each 71, point. Of p and no ,contains any point of P - . then evei \ point of IL, lies in a black (open) Voronoicell

Let U. l e the event that each B. contains at least one point of .

and no 8,,, contains a point of P For .r E B, , every point of B,strictly closet to than any point outside 13,. is Hence, whenever L.T,.

holds. eve' v Point E C U Be is black.The event ET, depends on the restriction of (P c , P- ) to the region R,c and E E(Za ) do not slime a vertex, then and RI are disjoint,

or meet only at boundat v points of both. We may ignore the probability

zeta event that the bulimia! V of any Re contains a point of P = Pi UP

Thus, Ur and U1 ate independent In fact. it S. T C E(27.,") ate sets of

edges at graph distance at least I how each other, then the sets Uicsand UrET 17, tneet onl y in t !wit boundaries, so the families {tic c E S}

279 Continuum pmrolation

and LT, C T lie i ndependent Thus. taking a bond o //I Di tobe open if mid oni U, holds we have defined a 1-independent bondpet colation model on Ed

The probability of the event U,. is

PA(Lr„) = — exp(—pAv4) exp(--(I — p) And),

whew nit is the volume of a [Ttlimensional ball with radius 1/-1 and

//,/ < 3 x Tl ed is the wItune of the union of the balls 13,2

Choosing A huge enough that 0 9Ac, / > IQ say.. and then 9 0 8639

'Thus. in the I-independent bond petcolation model .11 on 27 1 , each bondis open with plobabilitL at least 0 8639 Hence. hom the result of Balls-

tet BolloMis and Walters [20(15). Lemma IS of Chapter 3. with positive

limb/Minty the origin is in au infinite open path P in III lViten thisevent happens. tlw get/met/it path 11E 6 ( C P/i consists entirely

of black points. so the black component Ca of the origin is unbounded

Hence lift the chosen valves of p and A. 1l(` have

O(p) = it (C0 is unbounded) > (1

showing that p H = m i (d) p < I

hi the other ect ion. we shall compare tandom Volonoi maculation

with a 2-independent site percolation measure on L'': the argument is

due to Balistm. Bolkrbris and Quits [2005]

Lemma 10, For welt tl > 2 we ham p it (if)> 0..

Mont rid C II, be the In peicrube

and. lif t t > 0 let 10 denote the /-neighbourhood of Let S,he the event that sonic ball of (adios 1/6 with trentte 1-1, containsno point of P = U'PT and let U, he the event that sonar blackcomponent. (union of open Votor toi cells) meets hot H,, and a point ofthe boundat n Off/L I "1 of 11[ I ' '° Note that if Co is unbounded thenU, holds tot every r such that Co meets

8 ,f Random Voconoi percolation 273

Clearl y . the event 5, depends oak on the restriction of ('P". P - ) to

H;, I/6) Also. U1 . is the event that Hal e is ti black path in 11;, I/6) joining

0(11,(1/11 Hence U, depends onl y on the colon's of the points

x (of P:) ) with a E Hil l/(1) If S„ does not hold. then the closest point

of P to any E H;. 1/6) is within distance 1/3. so the colom of .v is

determined by the testi iction of (PUP - ) to /4. 1/2) Thus, U,,\ Sy.

and hence U, US,. = U (U,,\ S e ) is determined by the restriction of

(P+ . p-) to /1,'/2)

If ti E :SU ate such that ler - > 2 lot smile then the interiors

the sets HI, 1/2) and 11;,.112) are disjoint. Let (7 be the graph with ver tex

set in which e and a' ate adjacent if nu - icT < I for even- i. so

U is a (3` ) -f)-regular graph faking (' to he open if U S, holds.

we obtain a 2-independent site percolation model elf on G. As A —

and p — 0 with Ap — 0, we have PA 1 ,(U, U S,) — By Lenuna II

of Chapter 3. it follows that 'bete ate A > 0 and (I 0 q

The arguments above ale vet'. crude: we Mute included them only to

give an indication of the basic techniques needed to handle the long-

r ange dependence in random Nroionoi percolation Pot huge fr. home\ el.

it turns out that the method of Lemma 1(1 gives a bound that is not that

tat from the bulk: the towe l bound in the result of Brdister, 13ollokis

and Quas [2005] below was obtained in this way The upper bound is

much mote difficult

Theorem 11. If 4 et sufficiently huge, then the el Meal prob ab ility pH(d)

fin random Voronoi percolation on R d ,tatetfics

2 - ( 1 (9il log d)" i < p H (d) < ( 1 2 - ( / ‘71/ log&

mhate C Is an absolute constant.

We now tu t u to the main topic of this section, the critical probability

lot miion' Voronoi percolation in the plane If P is a Poisson process

in the plane. then with probabilit y I the associated Voronor tessellation

has three cells meeting at even' ve t tea: we shall assume this alwa ys holds

how now on Thus the graph U-p is a triangulation of the plane.. and

Eidel s lot implies that the ayetage (tepee of a vet tex of Up is 6

Thus. J on avetage', ap looks like the triangular lattice This suggests

that random Voionoi percolation in the plane mat- be similar to site

27,1 Continuum percolation

percolation on the triangular lattice Of course, critical probability isnot just a function of average degree For example, it is easy to see thatfm any (small) e > 0 and any (large) d, h > 3, there is a k- connectedlattice A with all degrees at least d such that p7;1(A) > 1 — ct-

The numerical experiments of Winter Feld, &liven and Davis 119811mentioned earlier suggested that p H (2) = 1/2 There is also a compellingmathematical reason for believing this: random Voronoi percolation inthe plane has a self-duality proper ty that implies analogues of Lemma 1and Corollary 3 of Chapter 3, the basic starting point for tire proof of theHart Theorem, and of Kesten's result that p7.1 (T) = pfr (T) Tr,1/2, where T is the triangular lattice (Theorem 8 of Chapter 5).

Let I?= [a, /4 [c, (II be a rectangle in t he plane By a black horizontalcrossing of R. nve mean a piecewise linear path P C 1? starting on theleft-hand side of H. and ending on the the right-Laud side, such thatevery point :r E Pis black, i e lies in the Voronoi cell of some E Pwith t: open; see Figure 11 We write H(R) 111,(1?) for the event that

Figure 11 In the figure err the left, 11(R) holds: the thick line is a blackhorizontal crossing of R. In the figure on the right, /1(R) does not hold

1? has a black hot izontal crossing Equivalently. 11(1?) is the event thatthe set of black points in I? has a component, meet ing both the left- andright-hand sides of I?

Ignoring probability zero events, the event .11(R) holds if and only ifthere is an open path P =n rr2et in Cp such that meets theleft-hand side of R, 1:;., meets the right-hand side, and, for 1 < i < tthe cells V„, and 17,„. , meet inside I?

write 11„(1?) for the event that I? has a white horizontal crossing,

8.3 Random Voranoi percolation 275

and Vi,(/?), VAR) for the events that I? has a black or white ve rtical

crossing, respectively

Lemma 12 * Let I? be a 'rectangle in R2 Then precisely one of the

events Ilb (R) and V (B) holds

Proof The proof is essentially the saute as that of the correspondingresult for bond percolation on Z2 (or site percolation on the triangu-lar lattice) Defining the colours of the points in 1? from the Poissonprocesses (P + ,P – ), colour the points outside I? as in Figure 12

Figure 12. the shading inside the (square) rectangle R. is flour the Poissonprocess 11 1 ,(R) holds if and only if the outer black regions rue joined by ablack path, and V. (I) holds if and only if the outer white regions are joinedby a white path. Tracing the in/Efface between black and while regions showsthat one of these events q 11;st hold

Let I be the inter face punk i e the set of all points that are bothblack and white (lie in the boundaries of both black and white regions)With probability 1„ the set I is a drawing in the plane of a graph in whichevery vertex has degree exactly 3, apart front Four vertices of degree 1

As in previous proofs of this type, considering a path component of the

276 Continuum pereolai ion

interface shows t hat one Of HI V?) awl (I?) must Ind If boll; hold,A5 can be drawn in the plane

Cotollaxy 13, If S 1 4, anti 9quarc x [g. Ii ( > O. Mc/2( 11 ( 3 )) = 1/2

Proof Dv Le1 1 1111a 1 2 . lb/ U < p < 1 we have Pp(llt,(5))+P i,(Viv(S)) =Flipping the states of all sites now open to closed or vice V(11S11. sethat lily ( Vw(S)) Flt-/)(11,(8)) Fioni the sin/ i n/env of the Poisson procuss ,= Ifi.„),(III,(8)) Thus. P),(//b(8))+IPI_,,(1-1-1,(5))=. 1Taking p = 1/2 the result follows 0

It ;eedinau 11997j plover' ;01 analogue of Co t 011 8 1 N' 13 rtniceining 'es-sential loops in 'widow \lo t onoi percolation on the projective planelie untuoks that p8 (2) = 1/2 In' notnett v consideurtions floweret,although Co/011m v 13 is t he 'uison wh y. p 8 (2) = 1/2. I his Elkin' oh-serration is even further how a woof than in t he case of bond percolationon Z2

Fin the Val ions lattices w hose et it ical ptobabilitv is known exactly,ant of the ninny plaids of the trainer 1Kesten Theorem can be adaptedwith out too n i nth work 17o/ /widow Votonoi percolation . the situationis different Some of the tools used to stud y percolation on lattices docal v met t o t his cont ext , but ot het s do not In pal ticitlat e isno ohs 101111 was to wink .Nlenshikov's I hern CM. and no direct equi snleut

of t he Rosso -Se% mow AVelsh Thiess is known. although, as we shallsee late!. ;t weak loin; of t he hatterresult has been plover!

Let us tarn w our main /11111 in t his section, a sketch of the proof t hat/1 11 (2) = 1/2 We star t w ith Hat rim's Lemma, t he first of the basic / esultsfor ordin a ry percolation Burt do extend to I he \Tot onoi setting slate

its analogue for Vor onoi per colat ion, we need t o define 'increasing ( 1 \ cuts'in t his context

We call an event, E defined in toms of t he Poisson processes (Pt P-)block-inctrusing, or simpl y increacing, if. fin ever } configuration =(Xt. XI') E and ere; configut at ion = (X.1). ) ivith C X1E

and Ai l AT, we have E In other words. E is preserved lw theaddit ion of open points, t he deletion of closed points 1 he definition ofan increasing function j (P + P-) is analogous: / is hue/easing if addingopen points or deleting closed points cannot decrease the value of .1

Since a point of 1R 2 is Hack if t he (a) newest point of p is open addingopen points and deleting closed points can change fhe colon/ of a point

S Random Voinnui ',ovulation 277

E R2 how vylire to black. but not vice versa any event defined

by the existence of cer brill black sets is incwasing: an example is the

event 1-1(R)= MVOIt is easy to see flow Hat is's Lemma that, with respect to P p . in-

cwasing events rue positively correlated, at least if they are sufficientlywell behaved In fact. all increasing events are positively correlated Tosee this, let us first ()onsider the case of a single Poisson process ofintensity I on 1?2 Let us wtite (Q 1 iF ) lot the corresponding proba-bility space, so Str is the set of all dist:tete subsets of F 2 . P i is thepr obabilit y Wen:it/In defining the Poisson process We shall write E i forthe associated expectation. Let i = I. 2 he two bounded measurablefunctions on 0.1 1 , P i 1. and suppose that each is in I lie sensethat ir) C u/ implies ,b (w) < (La')

Following Roy [1991) let E k he the a-field gene/rued hi the rolloir ingiulou ttation: set g = 2 k , divide Hu. a] 2 into do squares of side-lengthl/n, and decide fru each whether or not it contains points of Y EThus Er. pm (Rim 's Q., into 2 4 " pmts. with each part consisting of all

E C-2 1 that have at least one point in each of cei lain small squaws.and no points in certain other small squates Any two discrete sets lie in

differ ent pat Is of iEk lot large enough I. so t lie sequence Er. is a lint at ionof M t , Pi)

As E l (pr 1. ma y be viewed as an increasing flint( i011 On a discreteproduct space. bout Lemma -I of Chapter 2 (a simple coiolltu t of Hauis'sLemma) we have

EI(EW/i :0EM2 1:0)

(EMI E k)) E 01(1/4 I Ek))(*))

Ei(m)Ei(f/2)

Since g i is hounded mid Er. is a lint at ion, lir, the Mat ti le ConvergenceTheorem (see, [or example. 'Williams I1991D

E t ( g i I L k) 22' C/a

almost er el \ here. Hence la dominated comet gence. the left-hand sideof (5) converges to (g i go). proving t hat

Et ( g i q2) POE' 02) (6)

whenever th and gu are nwasumble hounded and Met easingThe cot tesponding result flu the colour ed process follows immediately

\Vt.) state this 418 a lemma giving onl y the character istic function case- although the proof Fin increasing functions is the smile we shall not

Conti truant percola lion

need the result. Hem. then is the equivalent of Harris's Lemma fortaudom Voronoi percolation in the plane Of course. a correspondingresult holds ha random Vol own percolation in Ea

Lemma 14. Let Ei = E i (P + .2 - ). = 1.2. be two black-Me

events. Then lo t any 0 < p < 1 use hone

IPy (E, n IP,,(E1)Pp(E2),

where iti the probability measure minor:toted to fandom Volonoi per-

colation in the plane

Pl oy/ -We shall w i ne E i e )Cei at ion with erespect. to ELp Let,

J; be the elm/atter istic function /if E h so h is black-increasing FixingF lot (guilt h. I (Pr .P - ) is increasing in P l.-. so IA O.

Eli 12 I P � P- HE(/ I P-4

fat every possible 2 - Taking the expectation of both sides ((wet Ps),

GUif2) ri(Et (r I P - )E( ig 2))

But as 1' 1 . Ei ale dee/easing in P , so are the functions g i = E(1; I P-).Applying (6) to 1 —y 1 sad 1 -go. which ate incteasing functions °LP-

( thl Y. we see tha t E g ig2) > E(gr )E(q) that

EP4 Tr P .-pa/H.21 P - 1) � E(E(li P-))1E(E(./2 P-))

= 11': ( )E( /2)

Combining I he last. two inequalities, we see that Et J i Li) > f i iEt (p),

as el/Mucci q

The uniqueness theorem of Aizumnan. Kesler ' and Newman [1087],Theorem 4 of Chaplet 5. also carries over to the Waonor setting (in anydimension). Just as fur the Gilbert model, the proof given by Buttonand Keane [1989] goes through. although a little cu t e is needed with thedetails: we shall not spell these out

Theorem 15. Fm any A > and any 0 < p < 1 thene is alumni ninetyat most one infinite open cluster in Op q

In Chaple t 5. we presented a proof of Hai / is's Theorem due to ZhangThis moo( retied on the basic crossing lemma (Lemma 1 Of Chapter 3).

8.d Random Voronoi percolatton 279

Harris's Lemma, and the symmetr y of the square lattice In his rut-published MSc thesis. Zva y itch [1996] pointed out that Zhang's proofcarries over to the random Voronoi setting, giving the following result.

Theorem 16. Fm random Voronoi percolation in the plane, 0(1/2) = 0Hence, pn(2) > 1/2. q

Unfortunately, none of the many proofs of Kesten's Theorem seemsto adapt to the random Voronoi setting: there seems to be no easy wayof proving the analogue of Kesten's Theorem Nevertheless, using anargument which is considerably more involved that any of the proofsof Kesten's Theorem, Bollolgis and Riordan [2006al did prove this ana-

logue

Theorem 17. Fm nindom Voronoi inoculation h i the plane, par = 1/2.

As we shall see at the end of this section, the proof of this result alsogives an analogue of Kesten's exponential decay result, 'Theorem 1.2 of

Chapter 3, and so establishes that the natural analogue of pr is alsoequal to 1/2

As /Mira/ Iced above, the wool of Theorem 17 is rather involved, so weshall describe only its key ideas: for full details we refer the reader tothe or iginal paper

Perhaps surprisingly, the lack of independence in the random Voronoimodel turns out not to be the main problem (or even one of the mainproblems) Indeed, it is easy to see that fig rectangles I? and say,separated by a moderate distance, any events E. E' depending only onthe colours of the points in B. and B.' respectively are almost indepen-dent In particular, there are independent events E. E' that almostcoincide with E and E': this property turns out to be just as good asindependence We make this idea more precise in the lemma belowHem, we think of s as the scale of the rectangles I? and 1?!: we shalltake s cc with all other parameters fixed From now on, we fix the

irrelevant scaling of the Poisson process 'P = TA by setting A =1

Given p > 1, s > 1 and a ps by s rectangle B, C R2 with anyorientation, we write F(Its ) = F,(R.,;) for the event that every ballB, (a:), E contains at least one point of P = P+ U where

= 2y/log s (It does not matter whether we consider open or dosed

balls; up to a probability zero event, which we shall ignore, this makes

no clifference )

28(1 Continuant Navarreton

Lemma 18. Let p> I be mistant let C IR2 he a /pi ha s rectangle,> 1 god set r 2v/logs Theo (R,) = 1",(R,,) holds with probability

— o(1) as s rilso if E(R s ) is any event defined 0 0 1 0 in termsof the colours of the points in R s . then E(R,)11F(R 1/2 ) depends only onthe rest/idiot) of IP -1 .P — ) to the -rag a hbourhood R,

Progl The firs t statement is immediate horn the properties of it Poissonprocess Indeed. we 1111W covet Rs with 0( s2 / I2 ) = s2 ) disjoin t (halfopen) scrum ps S, of side-length r [ n122 The area of each 9, is /2/22 hogs. so then number of points of P in has a Poisson dist) Haitianwith mean 210g s Hence. (he probability that a particular 8, containsII() points of P is exact Is C2 Thus. 1eitlr in obabilit \ I ) [ A vi v S.contains a point of P. and FL/11,1 holds

The second statement is immediate Itonn the definition of 17(13.4:11nrargument is as in the p i vot of Lemma 10 above Indeed E(R...) chantydepends oils on the restrict ion of (Pt P — ) to the -neighbour hood ./ef)of R, [(Ili's) holds then the closest point of P to �111V .r E R, is rpoint of B, (r), and hence a point of ri in Mc') Thus the colour of ,r isdet e i mined by the closest point of the restriction of (P I . P — ) to [((;)

Of course thew is nothing special about rectangles in Lemma 18;

choosing t he slim let site of the rectangle am 010 'scale parameter s willbe convenient lacer Also Lemma 18 has an analogue in am dimension,concei ving cuboids say

mi ning now to the specific stud y of dimension t \so. the lundiunentalquantit y that wee Awn!: %vitt; is the probabilik of a black crossing of arectangle. Let

/„(p, s) = (HO. ps] x [0, si))

he the P/3 -piobabilits that a ps bs s Ieelangle has a black Innizontalcrossing start with two easy observations about the behaviour ofji (p s) Fi t st

/p(m)s) 5 1,(112. s)

whenever p l < pg, as the cor i esponding events ale nested

'The second obsei sat ion is that

/ / ,(01 — ) / ),(m s )hAt)2. s )1),( s ) (7)

foi all m > 1. The argument is exactl y as in Chapter 3: Iet R 1 and

R 2 he m s ha s and /srs he s rectangles intersect lug in an s by s squaw

S Bondi n own porcolotion 98

Figure 13 1 vo rectangles Pr and lip intersecting in a square S. 11 P andP_ are Hack f t .arzontal crossings of fir arc P2 , respectively, an 1 Pr is a blackvertical crossing of then P t U U Pr contains a black horizontal crossingof Pt U Po

S. as in Figure 13 The events H(R I ). 11(R ” ) and F(S) ate increasing

Thus. la Ha t is s Lemma. in the Form of Lemma H.

PgH(R ni-R re2 )nits1) > Pp(H(RWPI(H(R2))11'1,(1.(3))

If 1-1(R I )na(R2 )n r (5) holds, then so does II (R U (see 1:',.igitte

1

As re,(1"(8))= P(H(8)) = s). I he telatimi (7) follows

The itatin al analogue of the Russo Seymour Welsh (RSW) 'Thewern

would state that if fp (1, s) > s > a then p(p. s) > ird > for

some function q(9, E) not depending on s (so. in particulin frit .(2. ․ ) is

bounded awa y flout zero) Such a statement: has not been proved lot

ardour Voionoi nett:oration

ft might seem that the proof of the RS \V-type thethent tee presented

in Chapter 3 would early me t to the random Voionoi setting. The

strut is indeed promising: thew is no problem defining the left-most

black vertical crossing' Lt (P) of R whenever 1 1 (R) holds: we simply

follow a black-white interface as befote. The problem is the next step:

the event LT(R) = P is independent of the stales of the points of

P to the light of P. but not of II/eh positions: we can find LP(R)

without looking at the colours of the cells to the light of 1,1"(R), hut

knowing LI ; (1?) tells tts whew tlw centres of these cells are, and that

thew ate no points of P in certain discs, puts of which are to the light

of .LII (R) Thus, there is no simple way of showing that the limitability

that LI:(R) is joined by a black path to the tight side of R is at. least

P(1-1(R)) Of course, a general problem wit h iandoin Voronoi percolation

is that no two regions ate independent, although well-separated legions

ate asymptotically independent

The first step towards the proof of Theotent 17 is the following result:.

although this is considerably weaker than a direct analogue of the RSW

Theorem. it tutus out to be sufficient lb/ the determination of pa (2)

282 Continirilin percolation

Theorem 19. Let 0 1 be fixed If lint inf, ( s)

0, then lint sup,_, f,,(p, ․)> 0

Lemma 12 states that the hypothesis of this themem is satisfied forp= 1/2: thus Theorem 19 has the following consequence

Corollary 20. Let p > 1 be fixed. There is a constant co = to(P) 0such that .for every so them is an s > sa with f il s(p, ․ )> Co.

The proof of Theorem 19 is rather lengthy, so we shall give onlysketch

Proof We proceed in two stages, throughout assuming for a contradie2tion that the result does not hold Thus, there is a constant cm > 0 suchthat

40, (8)

for all large enough s, but, for some fixed p > 1,

Ip(IL, s) 0 (9)

Here, and throughout, the limit is taken as s pc with all other pa-rameters, for example p. fixed. In this context, an event holds with high

probabititw or why), if it holds with probability 1 — o(1) as s cc.

'The assumptions above imply that. Mr any fixed E > 0,

fp (l +5,5) 0 (10)

as s xi Indeed. taking > (p —1)/5 and using (7) k times,

fp (p, ․ ) . n,(1 /,76, s) fff ,(1 s)fp(1, fp (1 (k — 1) 5 , s) >

(b,(1 + s) fp (1, s)) k f„(1, s) 71-i. 0,

contradicting (9)Condition (10) imposes very severe restrictions on the possible black

paths crossing a square, fOr example: roughly speaking, no segment ofsuch a path can cross horizontally a rectangle that is even slightl y longerthan high Thus, fot any E > 0, wisp all black horizontal crossings of agiven s by 5 squat e 5' pass within distance E s of the top and bottom of5': otherwise, with positive probability we could find a black horizontalcrossing of one of two 3 by (1 — E)s rectangles, contradicting (10)

The next observation is that wh ip any black horizontal crossing ofair s by s square 5' starts ar i d ends near the midpoints of the vertical

S 3 Random I lmonoi percolation 283

sides. Indeed, if there is a positive pr obability that some black crossing

P starts at least es/2 above the midpoint of the left-hand side, then,reflecting S in a horizontal line and shifting it vertically by es to obtain asquare there is also a positive probability that some black horizontalcrossing P' of 5' star ts at least 6512 below the middle height of 8', andhence below P. By Harris's Lemma, crossings P and P' with the statedproper ties then exist simultaneously with positive probability But, whip,P must pass within distance es/3, say, of the bottom of 8, and hence,below S' This forces P and P' to meet; see Figure 14 As P' passes

8'

Figure 14 Two squares S and S'; their horizontal axes of symmetry are shownbye clotted lines Each of the horizontal crossings P. P' of S and 5' passes nearthe top and bottom of the square it crosses. If P starts above P' then t hepaths cross: P ' must leave the region bouncied by part t of the bounds ' v of S'and the initial segment of P up to the point it

within distance Es/3 of the top of we find a black vertical crossingof a rectangle taller than it is wide Using rotational symmetry of themodel., this contradicts (10)

In a moment:, we shall state precisely a consequence of (10).. Fornow, we continue with our sketch of the argument Suppose that P isa black horizontal crossing of the square S = [O, x [0, s] Let Pr be

the initial segment of P, starting on the line x = 0 and stopping thefirst time we reach s = 0.99s Similarly, let I2 be the final segment ofP, obtained by tracing P backwards horn the = s until the firsttime we reach x = 0 01.s; see Figure 15. Arguments similar to thoseabove show that each of Pr and P, star is and ends yen \' close to the line

= s/2. Fur thermore, one can show that, whip, since each Pi crosses a

28-1 Continuum petrol& an

Figure 15 A squa t t S (outer lines) and a black path P crossing it (solid anddot ted curves) 'The path P i is the initial segment of P s:opr ing at Hie innervertical line The he al segment P. of P is defined similarh Ii ;uglily speaking,each Pi crosses a square smaller than 5', and so cannot a vroach too close tothe top and hot ton of .9 Thus ;24. P: her wise a slightly flattenedrectangle in S would have a black horizontal crossing T he dotted p i nt atin fact passes very close to the top mid botto m of S

rectangle of width II 99s, it remains within height ±(0 -195 4- 5)s of itsstinting points. It follows Hat Pr and A are wh ip disjoint: otherwise,P = Pr U is a Hack crossing of 5' that does not conic within height0 004s, say, of the top and hottom of S: the probability that such acrossing exists is o(1) by (10) But then each of Pr and P, similarlycontains two disjoint crossings of rectangles of width 0 98, and so on

ft is not hard to make the vague rugument just outlined pwcise, al-though it, turns out to be bet ter to work in a r (retail*, [0. x [—Cs. Ci],say, of bounded aspect ratio 2C > I. In Bollobris and Riordan [2006a1,Hie following consequence of (10) is proved

Claim 21. Let C > 0 he ficed and lel 1? = be the y by 2Cs rectangle

[0.s] x [—Cs.Csj Pot < j < 4. set [js1100.(1 90) ,4/1001 x[—Cs.Cs] Assam:My Mat (10) holds why eve, y black path. P (71),;sing

I? ha/C.:on/oily contains 16 disjoint black paths P. 1 i < aqierc

each Pi crosses ,/onte 1?) Ito/ icantally.

In other \t'olds, the path P contains 10 disjoint paths 1 i winding

S3 Random 1 /01'0110i percolation 285

backwards and to wards in a manner similar to the paths in Figure 15Of course, in proving Claim 21, one must: be a lit tle mo t e careful titan inthe vague outlirie above Never tireless, the proof is fair ly straightfor ward:it uses only- Harris's Lemma, certain symmetr ies of the nuclei, and (H))The last condition is applied to show that \\lip none of a fixed numberN = N(C.e) of rectangles with aspect ratio 1 + s is crossed the longvu}' In a black path

The conclusion of Claim 21 is clearly absurd: 1? has a black horizontalcrossing with probability at least f,(1, s), which is hounded away fromzero Taking the shortest black horizontal crossing of R. this somehowwinds backwards and forwards almost all the way across R. containingsegments that star t and end in almost t he save place lint somehow nevermeet each other Also, the shot test crossing of R is almost, certainly atleast 16 times longer than the shortest crossing of a slightl y narrowerrectangle As the length of a crossing of R, cannot scale as more than52 , this is impossible

All this sounds convincing: nevertheless. it is not so easy to deduce a

contradiction The problem is t hat, while the constant 16 in Claim 21can he replaced by any l arger constant it cannot be replaced by a func-tion of s tending to infinity: we can only appl y (10) directl y ton boundednumber of rectangles. Otherwise.. t he o(I) error probabilities il(t1/11111-

lateEn get around this, we use alutost-independence of disjoint regions to

.seurre the error probability' Tire idea is to consider the length (as apiecewise linear path) 1,(IR) of the shortest black horizontal crossing Pof certain .s by 2.9 rectangles R. when such a P exists We take L(1?,) =x it I?, has no black horizontal crossing. Even fen widely separatedrectangles. L(Rs ) and L(131,) are not independent, so we modif y thedefinition slightly- to achieve independence

As in Lemma 18. let P(R s ) = (Rs ) be the event that e ye/ y ball ofradius r = 2Vlog s centred at a point in R., contains at least one point.of P, and set

7(1?,) =L(R)F(Rs) holds

0 of her wise

13v Lemma 18. 1,(1?,) = 1,(R.) whp Furthermore. 1.(R,) depends onlyon the restriction of (P+ ,P-) to the t meighbom hood of 17, Hence. ifR s and 1r, me separated by a distance of at least 2r o(a). tire randomvariables 1,(17,) and Z (TC) are independent

Roughly speaking, we wish to relate t he distribution of i,(R J. which

286 Continuum pc/colahop

depends only an to the distribution of L(I? s /2 ), using Claim 21[fact to leave a lit t le elbow room, we relate L(I?„) to L( Bo ). say For71> 0 a (very small) constant, define g(s) by

g(s) = sup{ : CL- (R s ) < <

where I?, i s any s by 2s rectangle Recall that that L(1?,) < at if andonly if I?, has some black horizontal crossing, an event of probabilityf,,(1/2, 2s) > > cy > 0, horn (8) As L(R.,,) L,(1?,; ) Min, itfollows that 0 g(s) < cc if q is chosen small enough. and then s largeenough, which we shall assume from now on Also, the sumer/m i n aboveis attained. so

Pii(L(Rs) < 1/( s )) ( 11)

Claim 21 tells us that, whp, ever y black horizontal crossing of R.,[[including the shortest. contains 16 disjoint crossings P, of slightlyrower rectangles. Using the fact that whp every crossing of a 0 96 9 by2.5 rectangle is actually almost, a crossing of a square, we can place abounded number of 0 47s by 0 94s rectangles 1 < i < N, such that;w hp, each Pi includes crossings of some pair (R, R,;) of these rectangles,with and Bk, separated by a distance of at least 0 01.s Here N is anabsolute constant.. For each such pair (16,Ra), the variables L(B' ) andL(R k ) are independent, so

(1.1 (11 ) ) 7(&) < g(0 17s)) 5 Pi, (11(Rj ),11(Rk ) < q(0.47s))

= Fn (2:(R') < g ( 0117s )) illp (Z ( Rk) < g(0 47s)) 5 112

from (11) Hence, the probability that i(IL) ) + 2,;(1?k ) < g(0 .17s) holdsfor saute pair (Hi , Bk ) separated by distance 0 Ols is at roost N9 andhence at roost 11/2 if we choose ri small enough

The proof of !lemur 19 is essentially complete: horn Me remarksabove, whp every black horizontal ossing of B. has length at least16 times the minimum of L(R 1 ) + .L(Rk ) over separated pails (B1. Rk).

Thus,

L(R,) < 16g(0.47s)) g/2 o(1) < q,

so g(s) > 160 47s), and g(s) grows faster than s [1 , say As g(so) > 0 forsome so, it follows drat there are arbitrarily huge s with (3 3 < g(s) < cc.

But then. with probability bounded away horn 1, the shortest blackhor izontal crossing of IL, = [0, ,s1 x [0,2s1 has length at least [[; 3 'This is

$ 3 Random Voronoi percolation 287

impossible, since, will). R. meets only 0(s2 ) Vorouoi cells, and each hasdiameter at most 0(log s) q

As noted in the original paper, the proof of Theorem 19 outlinedabove uses rather few properties of random Voronoi percolation: certainsymmetries, the fact that horizontal and vertical crossings of a rectanglemust meet, and an asymptotic independence property. For this reason,the proof carries over to many other contexts; see, for example, Bollobrisand Riordan [2006b] \Tan den Berg, Brouwer and VrigvOlgyi [2006]proved a variant of this result for 'self-destructive percolation', a modelof forest growth taking into account forest fires, which they used to proveresults about the continuity of the percolation probability in this model

Using Corollary 20 and Harris's Lemma, Theorem 19 easily impliesthat 0(1/2) = 0, giving an alternative proof of Theorem 16 Of course,this result can be proved more cleanly by adapting Zhang's proof ofKestert's Theorem; see Zvavitch [1996] However, Corollary 20 is a goodstar ting point for the proof of the analogue of Kesten's Theorem, namelyTheorem 17

The main idea of the proof is to use some kind of sharp-thresholdresult to deduce that, for any p > 1/2 and any so, there is an > sowith

f„(3,5) > 0.99, (12)

say Before sketching the (rather lengthy) proof of (12), let us see howTheorem 17 follows The argument:, based on 1-independent bond per-colation on 2iL2 . is very close to an al gument given in Chapter 3

Given a 3s by s rectangle I?, with the long axis horizontal, let

G(R.„,) = II(Rs ) n V(5 1 ) n V(S2),

where S I and 5, are the two s by send squares' of R see Figure 16

Figure 16 A 3s by s rectangle such that G(R s ) holds

ji,(3, ․ ) > 0 99.

B y translational and rotational symmetr y.

Pp ( I ( .52)) = (Sr = p( ERS1 )) = (

288 Continuum purolation

so P 1,(0(.11„)) > 0,97 Let

G(Rh), G(R.,) n F(1?„),

and define elkfil ,C) similarly for a 33 by 5 rectangle RI, with the long sidevertical From Lemma IS, Pp (F(Rs)) 1 as s x Thus, if s is hugeenough,

P, (G(R, )) > 09

(1;3)

for every 3s by s rectangle Rs . By Lemma 18, the event 61 (R s ) de-pends only on the restriction of the Poissiu processes (P lP ,P-) to theTmeighbewa hood of B. Choosing s larg , enough. we may assume that

= 2Vlog < s/2

For each horizontal bond 0 = ((a, b). (a + 1, IR) of V let

= [gas, 2as + 3s] x 2b9+ s)

be a 3 3 by s rectangl e in 1E 2 Similxtly for e = ((a. b). (a,b 1)). set

,R, = 12as, 2as + x [21m. 2bs + 33]

Let us define a bond percolation model AI on 1 2 by taking a bond c tobe open if and only if G(R„) holds. If e and I' ale vertex-disjoint bondsof 12 , then the rectangles and RI are separated by a distance of atleast s. Hence. if S and T me sets of bonds of V at graph distance atleast 1, then the families {C(/?) : 0 E and {6(R„ ) : e E T} dependon the restrictions of the Poisson processes (P-4),P-) to the disjointregions MI) and UrE7 RI B t , and are thus independent Hence, the

probability mensme associated to AI is 1-independent From (13), eachbond is open in 21 with probability at least 0 9 so, from Lemma 15 ofChapter 3. with positive probability the origin is in an infinite open pathin A7 Such a path guarantees an infinite black path meeting [0. SP (seeFigure 11 of Chapter 3), so

1F),([0, s] = meets an unbounded black component) > 0.

The increasing event that even- point in [0. SP is black has positiveprobability, If this event holds and (0, meets an unbounded blackcomponent, then Co is unbounded From Harris's Lemma, this happenswith positive probability so 0( /)) > 0 As p > 1/2 was arbitrary, and0(1/2) = Theorem 17 Follow s

Our aim for the rest of the section is to sketch the proof of (12). thatlot any p > 1/2 and an) so thew is an s > 3 0 with f) ,(3.. 3 ) > 0.99: as we

Random l'oronot percolation 289

have seen, this implies 'Theorem IT As a starting point. Corollary 20is strong enough: one o l d\ needs / 1/0(3, s) > err for ma l e sufficiently

huge s In a discrete setting, we could easily use the Ftiedgut-kalaisharp-threshold result (Theorem 13 of Chapter . 2) to deduce that, for

this s, we have fp (3, s) > 0.99 It is not unreasonable to expect it to be asimple matter to adapt this argument to random Voronoi percolation, bychoosing a suitable discrete approximation, as in the proof of Theorem 3.say However, as we shall see, there is a problem

In Chapter if, we presented various methods of deducing the statement,fin bond percolation on V that corresponds to l i,(3, ․ ) 1 for p > 1/2

fixed One of these. the method based on the study of symmetric events..

originated in the context of random Voronoi percolation the others donot seem to adapt to this setting As in Chapter :3, to define symmetricevents we shall work in the torus, i.e.. the quotient of Pf 2 by a lattice

Mote precisely. we shall work irr the torus

72 = L o = R2/(10sZ)2

i.e the sox face obtained from the square (0. l(Is) x (0,10,s1 Ire identifying

opposite sides Here s will be out scale parameter; we shall considerrectangles R in 1'2 with side-lengths Afs„ where 1 < k < 9

The notion of a random Volonoi tessellation makes perfect sense onthe torus: we star t with Poisson pr ocesses P+ and P - of intensity p and

I -p on I'm which we may take to he the restriction of our processes onPi2 to (0. 10sI2 •P = UP- ate as berme. We shall always consider s huge It is easyto check that whp every disc in of radius \/log S. sa y, contains a pointof P, so the diameters of all Vinonoi cells are at most 2 Ylog s = o(s)

In particular., no cell f ivr aps mound' the torusThe rectangles R we consider also do not come close to wrapping

around the tot us Thus, events such as .1-1 (1?) have almost the same

probability whether we regard I? as a rectangle in R2 ca in T2:7:2

11= , (H(R)) = (I-I(R))+, o(I)

as s — where Pi, and Pr;' = F,, are the probability measures associ-

ated to Poisson processes (P lil .P-) on '12 = Trotand on R2 respectively

There is ji natural notion of a'symmetric' event with respect to themeasure P2„ namely, an event that is preserved by translations of thetorus The is one of t he two ingredients required for the application ofthe Ft iedgut-Kalai shat p-t lueshold result: the other is a discrete pr oduct

space

The definitions of the \Unman cells and of the graph Op.

a


Let 6 = 6(s) > 0 be such that 109/6 is an integer, and divide T 2 = Tiosup into (10s/6) 2 squares 51 of side-length S in the natural way Weshall approximate (Pt P-) (defined on the torus) with a finite productmeasure as follows: a square S i is bad if [Si n > 0, neutral if 1Si nP - 1 = 1Si fl P + 1 = 0, and good if 181 n 0 and n P41 > 0; seeFigure 17 Thus, open points (points of 7) ± ) ate good, and closed points

Figure 17 A small part of the decomposition of the torus into squares Sr.Points of P" are shown by dots those of P.- by crosses A square is good(heavily shaded) if it meets P -1 but not r , neutral (lightly shaded) if it meetsneither P 1- nor P-, and bad (unshaded) if it meets PM The 'crude state1. ofthe torus is given by the shading of the squares Usually, almost all squaresare neutral

are bad, and the presence of a bad point outweighs that of a good one;we shall see the reason for this choice later

For each 4, the probabilities that 51 is bad, neutral and good ate

it had - exp (-62 (1 - p)) 82 ( 1 — p),

Pnemrrat exp(-82) = 1 - 0(62 ), and ( 11)

Prood eXP ( -62 (i - p)) (1 - exp(-52p))

respectively, and these events are independent for different; squaresBy the trade stale of Si we mean whether Si is had, neutral, or good;the erode slate CS = C5,5 of the torus is given by the crude states ofall (10s/6) 2 squares Si

It is cleat that if S is sufficiently small as a function of s, then CSessentially determines (I) + , P-), and thus Op (defined on the torus).

8.3 Random 1/monai percolation. 291

Indeed, for s fixed, the a-fields generated by CS =CSo are a. filtration72

of that associated to PI, In fact, fen the discrete approximation to (es-sentially) encode the entire graph Gp, it is enough to take 6 = a(811).

Indeed, it is easy to check that if S is this small, then two things happenwisp First, no Si contains more than one point of P, so CS essentiallydetermines (p' -1 ,P– ), up to shifting each point by a distance Sec-ond, there is no point of T2 such that the distances to the four closestpoints of P lie in an inter val 2 0.,(5] Thus, shifting points of Pby at most \FM does not change which Voronoi cells meet, and so doesriot affect the graph structure of Op

As we shall see. we shall not be able to take S this small, so the graphstructure of Op will be affected by the discrete approximation, althoughonly 'slightly'

The possible crude states CS of the torus correspond to the elementsof Q" = (-1,0,11", where a = (10s/S) 2 is the number of squares Si,

and for w (w 1 ) E Q" we take (+4 to be –1, 0 or 1 if Si is bad, neutralor good, respectively. Let us write Pp_ r,. for the product probabilitymeasure on Q" in which each w i is –1, 0 or 1 with respective probabilities1,–, 1 – p_ –p+ , anti pi. Then the discrete approximation to P. givenby C'S corresponds to the measure on Q" More precisely, ifE is any event that depends only on the crude state of the torus. then

P„ (E)= (F), (15)

where F C Q." is the corresponding event. and pg„„d and p arerelated by (11)

It will clearly be convenient to use a form of the Fr iedgut–Kalai sharp-threshold result for powers of a 3-element probability space (-1,0,11In this context, an event E C .C2" = (-1, 0, 1}" is increasing wheneverw (w i ) E .E and Ji > for every then w' = (c.,1) E E An eventE C Q" is symmetric if it is invariant under the action on 4" of somegroup acting transitively on [rd,-- {1,2, . .

From (1-1), both p kid and pg,„Ki will be very small, so the sharp-threshold result we shall need is an equivalent: of Theorem LI of Chap-ter 2 As noted by Bollobiis and Riordan [2006a], the proof given by

iedgut and Kalai [1996] extends immediately to give the following re-sult:

Theorem 22– There is fai absolute constant Ca such that, '11' < q_ <

p_ < 1/c, 0 < p+ < q+ < 1/e, E C 0, 11" is symmetric and

292 Contiaaoat ',cicalaI ion

increasing and Ft : (E) > then (E) > 1 —e whenever

win { ri+ — p_ — q- } tid log( 1/E)p„,„, log( 1 )/ log tt. ( to)

where p„„, nrx{g+.p_}.

When we come to apply this result, the hunt of (16) will /natter; thisis in sharp contrast to the situation lot bond percolation on 2: 2 , Indeed,when we used Thement 13 of Chapter 2 to (move Kestells 'Theorem inChaplet 3, the hun t of the era responding bound was not it/tiro/tautAll that, mattered was that, with s: fixed and II tending to infinity, theinctease in p required to taise P,,(E) front E to 1 — E tends to zeroMete even though we are using a strange/ tesult, (with the extra factorp,„„, log(.I/p,„;,„) working in out layout). flame is a limit as to how' smallwe can take (5 while still getting a useful result front Theorem 22

Vie shall eta/pant (E) with Pi t,(E) fot a suitable event E. when/p > 1/2 is fixed. Row Corollary 20„. the hate/ probability will be atleast some small constant E. and to prove (12) it will suffice to show that

(E) 1 — E. In doing this, we shall make a discrete approximationas above. Et ow (1+1). when we apply Theorem 22, we shall have

— p_ — a_ ,= i tS 2 — 6 2 /2 = e(a-)

Also, a will he constant. p„„, = 8(e5 2 ), and n = (10s/(5) 2 = e(r2/8.2)Substihdiug these quantities into (16)„ we see t hat this condition w ill besatisfied if and out- if a > lot some constant > 0 depending on pAs p ant/ °aches 1/2 front above. the constant ^1 tends to zero Thus, wecannot afford to take a lei ibly fine discrete approximation; recall that

= 0(S - ) vonld be needed for the alma oxiination (access not to affect,Op at all

With for - a 50 1 8 11 positive constant, out discretizationpottiest; will inhoduce ! defects' Mime, given the 'rounded' positions oft he points of P. we do not know which Voionoi cells actually meet Thedensity of these defects will be small (a negative power of s)„ so onemight expect them not to matte!. 'r Ids tutus out to he the case, but

n eeds a lot of work lo proveBenjamin' and Seta anon 11998] env/untie/ ed the same p i oblent in ram -

ing a certain 'confoi mal oval fiance' pi upc/ tv of random Vet/m ini per co-

lation in two and since dimensions This is not conformal inwat lancein the sense of the conjecture of skizentrunt and Langlands Pouliot and

Saint-Aubin [199-f] discussed in (Maple' 7 Instead, in two dimensions.,what Idenjamini and Scht alma proved is essentially the following Let

8 d Random Voronoi percolation 293

C IP1.2 he a (nice) domain. and let S i and 5"g be t g o segments of its

boundm y Consider t he Vol onoi percolation associated to a Poisson po t-

cess O f intensity A on 1?. using a cot tarn metric ds to form the Voronoi

cells, rather than tine usual Euclidean metric. Then. as A — a fixedconfor 'nal (locally angle-preset ving) change in the met:tic ds does notchange the probability that there is a black path horn S i to So by tame

than o(1)

Let us note that this is also a statement about detects: each \Towingcell is very small. and, locally, the change in the metric is multiplicationhe a constant factor, so there \\ill be very few 'defects' whore different.Vor (anti cells meet for the two metrics: in two dimensions the expectednumber of defects is bounded as A — in three dimensions there

are more but the density is still vei l: low the result of Benjamini

turd Schr arum is that these defects do not affect the crossing probabilitysignificantly: even though there ate very few defects the proof is far

nom simpleIn proving Theorem 17 we have a large] densit y of defects: on the

of het hand. we have the freedom to vary the probability p, replacing an

err bitril/ V p> 1/2 by (p+ 1/2)/2, say. Roughl n speaking. we shall show

that the effect of the defects is smaller than the effect of decreasing p in

ti l ls way

Let us nog make some of the above ideas precise Let 1? he a rectangle

in , 2 NVe wish to define an event E depending on the Poisson processes

(Pc . P-) on 72 :Mtn that, whenever E holds. then 11(1?) holds even if

we move the points of P a little. Given rt > 0. rte say that a point At E I"

is 1i -robustly blink if them is E P + with dist (.r. z') > dist(1 . , z)

for all o f E P-; in other words, the nearest open point of P is at least

a distance closer than Ow neatest closed point If rte move all points

of Pat most a distance if/2. then any .c E 1 2 that was if-robustly black

(and hence black) will still he black We SUN that a path P C 12 is

it-robustly Huck if every point of P is tfa ()busily black.

It tin US out that if 0 < pr < po < 1 and > 0, then we can couple

the probability measures Ft.), and so that, whp, for event' path PI

that is black with respect to the first measure, there is a 'nearby' path

Pr that is i f-robustly black with respect to the second measure, whew

=

Theorem 23, Let 0 < < and > 0 be given. Let ill =

u(s) be any function with q(s) < 111c way cough nil in the same

294 Con tinituill percolation

probability space Poisson processes P r* PT and o re/ = nos

with intensities p i! 1 – and 1 – po respectively, such that P;.4"and pi are independent foi each and Mt; following global event Eg

holds whp as s x foi every piecewisefinew path P 1 that is black withrespect to (Pt , pi ) there is a pieceivise-lineal path P that is 4h-robustly

black with respect to (PT ), with (In (Pr, Po) < (logs )2 ,

The proof of Theorem 23 is perhaps the hardest: part of the proofof Theorem 17, and we shall say very little about it Essentially, weroust deal with certain 'potential defects', where the four closest pointsof Pr = U' I– to some point E 11.2 are at almost the same distance(the distances differ In at roost s –e ( I )) We call such points of Pr bad,Roughh speaking, a bad cluster is a component in the graph on the hadpoints in which two bad points are adjacent if they are close enough tointeract. The heart of the proof of Theorem 23 is a proof that, tinder theconditions of the theorem, whp the largest bad cluster has size ° (log s).

As the probability that a given point is bad is 5 –e(0 , one might expectthe largest bad cluster to have size 0(1); it turns out. however, thatthere will be bad clitsters of size e(log s/ log log .․) Hence, the o(logs)bound needed for the proof of Theorem 23 is not far from best possible.

Using Theorems 22 and 23. it is not too hard to deduce (12) fromCol olltuv 20. and so prove 'Theorem 17..

Proof of Theo:T.7n 17 Let p > 1/2 be fixed As shown above, to provethat 0(p) > 0, it suffices to show that, given so, there is an s > so forwhich (12) holds

Let 7 be a positive constant to be chosen later In fact, we shall set= (p – 1/2)/C, where C is absolute constant Let er > so be a large

constant. such that all statements ' if .9 is large enough' in the restof the proof hold kw all s > sr 13), Corollary 20, there is an s >such that / 1/ .,(9, > co, where co > 0 is an absolute constant; we fixsuch an s throughout the proof. Let Rr be a fixed Os by 9 rectanglein P.,= , so P i p(11(R 1 )) > co. Regarding R i as a rectangle in the torusII" = let E 1 be the event that R i has a black horizontal crossingin the random Vcnonoi tessellation on the torus As noted earlier, sinceI?, does not come close to (within distance ()(log s) of, say) wrappingmound T2 , we have

a( I%i) = Pi';2(H(Ri)) o(1)

8 ,9 Random Volonoi percolation 295

as s oc. . In particular, if s is sufficiently large, we have

r;9(Er) .� co/2,

(17)

say.

Let 6 be chosen so that 10.9/6 is an integer, and S -2-1 < 6 < s14;

this is possible if s is huge enough, which we /nay enforce by our choiceof sr. We shall first apply Theorem 23, and then 'cliscretize' by dividingthe torus ¶2 into (108/5) 2 small squares S i of side-length 6, as above

Set p' = (p + 1/2)/2, so 1/2 < p < p, and consider the coupledPoisson processes (Pif , Pi) and (Pt PT ) whose existence is guaranteedby Theorem 23. applied with m = 1/2, po = p', and q =16 < c1

Let R. , be an Ss by 2s rectangle obtained by moving the vertical sidesof outwards by a distance 8/2, and the horizontal sides inwards bythe same distance; see Figure 18 Let E2 be the event that there is ahorizontal crossing of Ro that is IS-robustly black.

Es,

Figure 18 The 9s by s and Ss by 2s rectangles A and (riot to scale),together with a black path PI (solid curve) crossing R I horizontally anda nearby robustly black path P2 (dashed curve), part of which crosses kihorizontally

Suppose that Er holds with respect to (Pi", Pr); flour (17), this eventhas probability at least co /2 Let Pr C R 1 C T2 be a black path thatcrosses R I horizontally If the global event Eg described in Theorem 23

holds, as it does whim then there is a path A with (In (Pr P2 ) < (logs)'such that P, is IS-robustly black With respect to (Pt. If s issufficiently large that (log 8) 2 < s/2, then any such path 12, containsa sub-path A crossing horizontally, so E, holds with respect to(P:t,P.r) If s is large enough, then Eg holds with probability at least1 — e0 /4, so

(E2 ) > (E, ) — P(Eg ") > co/2 — co/4 > co/I

be the event that. some Ss by 2s rectangle it/ ¶2 has a 46-robustly

296 Continuum imirolati OP

black horizontal crossing Then

(E.)> P-, (Eo) > col

Let us divide T2 into n = (10s/6) squares Si as before, and definethe crude state of each Si and the etude state CS of the whole torusas above. Let :p; be the event that the crude state CS of the torus iscompatible with E3. Since P(E 3 ) = E(P(E3 CS)) < IF(ED, we lune

P;,, (EEO > co/4

Let ',brit' and proud be defined by (14), and let and p'gt. be definedsimilarly, but with p' in place of p. Thus.

=

//gond =

1 - exp (-62 (1 - p ')) ti S2 (.I - if). and

exp (-82 (1 - p'))(1 - exp(- 62 1/ )) (52p

As the event E:63 depends only on CS, it may be viewed as an event E3in the finite product space Q" = {-1, 0,1}" In particular, nom (15)

= P7; (E753 ) > co/4. (19)

The event F3 is symmetric, as E3 and hence El si is preserved by a trans-lation of the torus through (18, Pi. ), ) E Z. ft is not hard to check that1:3 is also increasing

From (11) and (18) we have

pit „ i (1-p1 )52 , pig,„„ 1 t-pe, /Thad ti (1 -p)82 . and it, p82 (2(1)

WC shall apply Theorem 22 to -1, 0, 11", n = (10s/8) 2 , with p_ =

= peso" , a_ = pima and = p800 , 1 , and with e = min{c0 /4, ill- In,say Note that z > 0 is an absolute constant

Let A be the quantity appearing on the light-hand side of (16) withthis choice of parameters From (20), all fern of the quantities p_ and

are at most if s is large enough Hence, p„,„, < 62 , and

A = 0(62 log(1/62 )/ loge).

where the implicit constant is absolute By om choice of 6, we have 1/6 <s2-f On the other hand, = (10s/8) 2 > s2 Thus, log(1/5 2 )/ log a < 27.Hence. A < Chi ,i 2 for some absolute constant C

As C is an absolute constant. we may go back and choose .7 smallenough that CT, < (p - p')/2. so

A 5_ (P - P/)62/2

8 (( Random Vownoi percolation 997

hot (20). we have

q+ - v+, P- ^" (1)- 91)52

Thus, if s is large enough, q+ -p+ and p_ - q_ ale larger. than A, i.e.,the hypotheses of Theorem 22 are satisfied Theorem 22 and (19) thenimply

(PII) I — 5 > 1 — 10 — WO

Retruning to the continuous setting, but considering the event Er),which depends only on CS, we have

p (E) = p F ) > - (")

Suppose that CS is such that LI holds, and let (P P-) be aqy con-figuration of the Poisson process consistent with this particular eludestate of the tot us By the definition of E7)') . there is another realizationconsistent with the same crude state such that E. holds. i e such thatthere is a 4S-robustly black path P Grossing some Ss be 2s rectangle in72 Prom the definition of bad, neutral and good squares, it follows thatin the realization (Pt P - ). the path P is still black Hence. -)\ riling E4

for the event that some Ss by 2s rectangle in T2 has a black horizontalCl owing. we have

PI, (Er) � 1Pi, (Etr ) > 1 - I"

The rest of t he argument is as in Chapter. 3: we can cover 72 with(is be -Is rectangles /?, , ,R9 5 so that, whenever Er holds, one of theR i has a black Mu izontal crossing Using the 'rah-root trick', it Followsthat

(H(R ) ))> 1 -

Returning to the plane,

fir (3/2,45) = (1-1(R1))+ o(1)> 0 999

s is large enough Appealing to (7), it is eas y to deduce that fp (3, -Is) >0 99. As noted earlier, 0(p) > (1 then follows easily by considering 1-independent percolation Since p > 1/2 was at bitranr, and 0(1/2) = 0from Theorem 16, the result follows q

The argument: above was just a sketch of the proof of Theotem 17hi particular. we said almost nothing about the moor of Theorem 23Although purely 'technical', such a result seems rather difficult to ploverthis is actually a huge part of the work in the or iginal paper

As shown by Bollobiis and Riordan [2006a], for p < 1/2, the proof of'Thoth ern 17 also gives exponential decay of the size of the open clusterCo or cot responding black cluster Crg containing the origin; this resultfollows from (12) by considering a suitable k-independent site percolationmodel on Z2

Theorem 24. Let leol denote the diameter of Co, the area of Co, orthe 711110101 ICT F of I/010710i cells in Co Then, for any p < 1/2 > there

is a constant c(p) > 0 such that

Pp (FCol n) exp(—e(p)n)

hi every n > 1 q

As in Chapters 3 and 4, this implies that the Hanunersley and Temper leycritical probabilities coincide

Theorem 25. Let p i he the Tempe/ley cr itical probability for random

Ijoronoi percolation in the plane above which Ej,(Kup diverges, where

roi denotes the diatnetei of Co, the area of Co, or the number of Voronai

crffs in Co Then. = p i( = 1/2 q

We have seen that :random Voronoi percolation can be very difficultto work with. Nevertheless, there is some hope that the conformalinvariance conjecture of Ai-tern/mu and Langlands, Pouliot and Saint-Aubin [1994] discussed in Chapter 7 may be approachable for this model:the model has much [note built-in syninteti y than site or bond percola-tion on ain lattice. Also, the result of Ben] amini and Schramm [1998]provides a possible alternative method of attack: roughly speaking, t heyshowed that for :random Voionoi percolation, conformal invariance isequivalent to 'density invariance', i e , the statement that, with suitablescaling, the crossing probabilities associated to two different inhornoge-neous Poisson processes converge

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Index

2-close domains, 211fl-regular infinite tree, 6:Farm exponents, 238k-branching tree, 6k-independent percolation, 70, 73, 108,

162, 250, 270, 287nth-root trick, 41

Aizelurian-Langlamils-Pouliot-Saint-Atibin conjecture, 178,181

amenable graph, 121Archinieclean lattice, 154, 155aspect ratio, 184attachment vertices/sites, 10, 157automorphism-invariant event, 1IS

ball (in a graph), 108Vat) den Berg--Nesten conjecture, 44Van den Berg-Kesten inequality, -12

in percolation, 78Bettie lattice, 6black cluster, 269black horizontal crossing, 274black-increasing event, 276bond percolation, 1bond, i e , edge, 1Boolean niodel, 240, 242box product, 42branding process, 8

cactus, 10Cardy's formula, 184Carleson's form of Cardy's formulacell percolation, 267closed bond/site, 1Colour-Switching Lemma,196configuration, 2, 268conformal

equivalence, 182

invariance, 176 239map, 178rectangle, 184

connective constant, 17continuum percolation, 240correlation inequalities

of Ahlswede and Boykin, 45of van den Berg and kesten, 43of Fortnin, liasteleyn and Cinibre, 6of Harris, 39of Reimer, 44

correlation length, 234coverage questions, 261critical area (Gilbert's model), 244critical degree (Gilbert's model), 24-1critical exponents, 62, 232, 234

for the triangular lattice, 236critical probability, 5cross-ratio, 183cube, 38

decreasingevent, 39set system, 39

dependent percolation, seek-independent percolation

directed graph, 24Dirichlet diagram/domain/polygon/

tessellation, see Voronoi tessellationdiscrete contour integral, 207discrete domain, 191

mesh of, 191discrete torus, 65

86 discrete triangular contour, 207domain in 12, 178clown-set, 39dual bond, 12dual graph, 12dual lattice, 12dual site, 12

320 Meted,

edge, sec bondequivalent

bonds 138percolation measures 81random graphs, 119sites. 1 85, 138weighted graphs, 156

exponential deep, 70 . 10-1for Gilberts [node/ 251for Voronoi percolation, 298

extended KagoutO lattice 155,external boundary

in IC, 216in :2 13

Nlenshikov, 93Newman and Schulman 1211Smirnov, 196

line graph, 3log-nioncit one, -16log-sitpermodular,

marked discrete domain,193/narked domain. /79Martingale Convergence Theoremmatching lattices, 146mixed percolation, 21, 24monochromatic path, 191multi-graph, 2

fact' percolation, 12, 2- 67 Newman Ziff method 175finite-type graph 85r.K.c; inequality, -61, 16 open bond/site, 1

open cluster, 3, 269Gilbert (disc) model, 2-10, 2-12 open dual cycle 16

open horizontal glossing 51Harris's Lemnm open our-cluster . 25, 111

for \Orono! percolation 278 open path, 3binusdor tf distance, 211 open subgraph, L 80Hex, game of, 130 open vertical crossing, 51hexagonal lattice. 9, 136, 155 oriented graph.. 2-1

bond percolation,1A9 oriented line-graph 82face percolation 131, 2-15 oriented path, 25site percolation, 1(i5 oriented percoIat ion, 21 , 167

hitting I adius, 256 out-ball. /0-1honeycomb lattice, see hezagonal lattice out-class 81horizontal dual 51 out-class graph, 8-1hype/cube 38 out-lilac sites 8-1hyperscaling relations 235 out-sphere. Sli

out-subgraph 25, 81increasing outer boundary. 192

event, 39 276function on a poser. partially (mimed set :38set system, 39 Peierls argument., 16

independent percolation measure, 3 percolation measure, 3, 80influence percolation probability, 5

of a amid in percolation 63 pivotalof a variable, /16 bond in percolation, 63

inner boundary 192 variable 16interface graph, 5-1 131 Ill-I, 275 planar lattice, 138

Poisson process, 2,11Jordan (10111:1111, 17!) Poisson-Voronoi tessellation me

ranc:an Voronoi tessellationKagons1 lattice, 15-1, 155 157 poset, 38Kohnogorcw's 0-1 law 36 power set, 37

Lemma: radius of a clusterBalkier Bollolnis mid Wallet -1 in an oriented graph, 86de Binijn anal Ercilis 37 random convex hull problem. 262Fekete 36 random geometric graphs. 2-10 25 -I11in ris 311 random set process_ 176Afar gulls and Russo .16 random Vinonid percolation 263

Inded: 321

pindout Voronoi tessellation, 263retticamalization argument, 68itliemann Mapping Theorem, 178robustl y black path, 293B;e1W T'lleorem, see

Russo-Seymour-Welsh T'hcoret

scale, scale parameter, 279scaling relations, 235Schramm-Loom/et . evolution (STE),

232, 235, 239self-avoiding walk, 15self-duality, 12, 129, 274sharp threshold, 49site percolation 1site. i e vertex ISLE. sae Schramm-Locuetner evolutionsphere (in a graph), 79square lattice, 2

bond percolation, 50site percolation, 133, 162 . 1165

square product, 42square-root trick, 41star-delta transformation, 148. 151stn--triangle transformation, sac

star-delta I ransforma I ionstale of a bond/site, 1stochastic domination, /57strongly connected graph, 26. 85sub-exponential decay, 76substitution method. 149, 156

Gilbert 246Grimmett 152Grinunett ancl StaceHall 246, 249Hammersley, 19, 106Harris 61, 124Harris and Kesten, 50, 13, 128 146Kahn, Nalai and Linial,Kesten, 67, 71, 131, 1.17Kohnogorov, 36Liggett., 171, 172Wester and Roy, 252

lienshiletut, 90. 96, 100Penrose, 255, 257Reimer, 44Roy 251Russo, 134Russo and Seymour and Welsh 57Smirnov, 187, 188StIlit'llOV and Werner, 236\ Vierrnan, 149, 169, 162Zravitch, 279

'Ft/lessen polygonalizat ion sac VoronoiIessellatiem

t ranslat ion-invariant event 120triangular lattice, 13, 136, 155

bond percolation, 1,19Grit ical exponents. 236site percolation, 129. 187

two-dimensional lattice, 182

symmetric uniqueness of the infinite/ cluster,event, 66 for (filbert's model, 252plane graph 138 for random Voronoi percolat ion, 278set system, 49 in amenable graphs, 121

up-set 39t ail event 36Theorem:

Aldswede and Daykin, 45AiZe11111811 and Barsky, 103Aizettman, Nester and Newman, 121Mammon and Newman, 107, 111Arzelit tend Ascoli, 227Balister, Bollobeis. Sarkar tend

Walters, 260Batiste! . Bollobas tend Stacey, 171Batiste') Bollokis and Quas, 273van den Berg tend Kest en, 43Bollobas and Riordan, 279, 282, 298Bourgain, Kahn, Katznelson

and 48Cara/ Intodory, 182Tot tuin ICasteleyn and °Milne, 46Friedgut and halal, 48, 49

vertex see sitevertical dual, 51Voronoi

cell, 263diagram, 265percolation, see random Voronoi

percolat ionpolygon, 265/ essellat ion, 263

weighted graph, 138 156weighted hypercube, 37weighted platuu hit tic( 138witness, 42

ill percolation, 78

Zill s foomila 186

List of notation

1,41: cardinality,ALB: symmetric difference, 38A 0 .41, A 08: square/box product., -12:1 C B: subset (equality allowed), 38A P.L : closed 6-/leighbourhood, 211A L : boundary are of a (discrete)

domain, 179, 193At: outer boundary arc of a discrete

domain, 19313i (a binomial distribution, 288,4,1:4 Indl in a graph, 108Br(z): ball in Ia 2 or C, L83Bh:D: out-ball in an oriented graph,

104C: complex number-s, 178

= {/1 E A : — y}: open clustercontaining :r, 4

G;f: open out-cluster, 25C'-: out-class graph, 8-1Dr y: occupied set in 0, , A 212El, etc: expectation associated to Pp

elm. 6B(A): edge set of a graph A, 1E15.(:): existence of open separating

path, 201E, A: vacant set in GI ,. A, 2430(o): 0, ,A with ers 2 A = 244G A (Il !): generalized Gilbert model, 2-13Cis: discrete domain, 191CTT: discrete approxinuttions to a

domain, 212G,„,7 : finite random geometric graph

with degree d, 255G, A: Gilbert model, 2-12G,: Get , 212C r (1,), (1 ' A): finite noulom

geometric graphs 254H: hexagonal lattice, 136

11(11)= 11 1,(R): 17 Ili”; a blackhorizontal crossing, 274

H„ lo-nearest graph, 257there me k infinite open clusters.

118P = PA: Poisson process, 2-11P A : open points of P, 268P closed points of P, 268

p : probability ineastit • e associated tobond percolation, 2

P';\probability measure associated tosite percolation, 2

(2, 11, (4: weighted hypercube, 37(2: mulch: ant of 27 2 16817,, (:r) = (4 + -1): there is an open path

from 9i (:r) to St (:r), 87S LEN : Schrtunin-Loewnet• evolution,

2355,(4): sphere in a graph, 70St (r): out-sphere in an oriented graph

86I: triangular lattice, 129:2 7 2 * torus, 954:2,, : 2 : discrete torus, 651 - (A): vertex set of a graph A, 1VI (11), 1- -E (R): R. has a black/white

vertical crossing, 2751:r : Voronoi cell of a, 263

natural orientation of 'Si / 10

(11(!): degree of a vertex/site 24d(r,y): graph distance, (iid(::, y) Um an oriented graph: distance

from a to Li, 86dist(x,y): Euclidean distance-, 211

flauselorff distance, 211f/z): probability of EA (:), 207

List of 17 o I ati o n 323

f p (p, .9 ): pa by s crossing probability,280

(4: continuous extension of f i}, 225It p (rn. n): In by n crossing probability

in 52 , 591,(m, n): h rr(nr,n), 59pH: lialffillettiley critical probability, 5PH (1): critical probability for Voronoi

percolation in Rd , 269pi : lemperley critical probability, 6pc : p H or pi , 9r (ex ): radius of open cluster, 61

in an oriented graph, 86s: scale paranteter, 279

— or {:r — y}: t here is an openpath front x to p, 3

{:r -L }: there is a long open path from:r, 79., 8(i

:r` — y}: uu open path front St (r) to(a : WI

{a rt' IL} = R,,(a): here is MI open pathGen Sj (t) to St (r), 87

A: a (usually infinite) graph, 1A': dual graph, 12

(A, p): a weighted graph, 138A li:: open subgraph in bond percolation,

A-;,: open subgraph in site percolation, 2

TV: oriented graph, 24out. subgraph, 25

influence, 46SA: resealed lattice. 179511: resealed hexagonal lattice, 191ST: resealed triaiiguha lattice, 187ll(p) = 0}; (p), etc: percolatirm

probability, 4x(p) = x!,.!(p) etc: expected cluster size,

6A( 4 ): connective constant, 17p,,(p): min x pn(a7 , p ), 88p11(x,p): probability of Rar(x), 87

(CO: inner boundary, 192(0): outer boundary. 192

it': external boundary. 13 216D : discrete contour integral. 207

a s: with probability I, 240whp: wit h probability I — of1i 240

bela bollobás, oliver riordan-percolation -cambridge university press (2006)

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