Today's planMultichordal SLEok 0 Rational function Shapiro's conjectureLDP Multichordal energy

LDP of SLE k soo E ERadial Launer chainBowsher Varadkar's theoremLeeuw Kufarer energy

Peltola

ttttxE.it

OO O

Definition Theorem CBeHara Petula WuFor oink EY the multi chordal SLEin Xy D X i Xzu is the

unique measure on multichords Hi DhXxl D Xa Xm Such that

Tj is the single SLEu in Dji simply c domain

CXy CD Xi y XenyD

rjplink pattern of ur points

y D e v

g x L fit Nn disjointX

connectingX i Xun

Dj accordingto 2

By Radon Nikodymderivative

I assume that Ock 85

K e 3k 8 lb K o2k

central charge of SLEK

dPak emultichordal seen

dipki indl t ith

a

exp l miser ik

Ekind expectedmoth Fnl

where my era Tn

pZzpijplllenri I.frof y

atleast pchords

Brownian loop measure

Lawler Schram Werner

conformity invariant

K so not for k o

Dcf It is natural to define MultidordedSLEo in Xy D x Xan

Ybif.in fgdinxa

eµ 5 la c hype's.ge

U

uniqueµ

mini miter of 1 Algebraic methodthe anltrchordalLearner energy

Results jointly with E Peltola

Them n Let Cy yn be a

geodesic multichord in Hf connecting

Xi Xyz EIR don't fix 2 yetThen there exists a rational funitionhey Pa E polynomials g

of degree netsI I

met preinage ofa regular ratesuch that ef h

y Npvun v yiu yi unit u R

hj f R n

Moreover hy has critical points atXi i Xzn J all

T.xdx.a.in7ia

YU V ko is critical part

of h of index k

if hcxj hsxoje.xxykcfgii

I hin.is ii tx

tx slit

by I I leg n

g

Classifying g multichord Rational fmhonwith critical pointsX Xm

try 13 unique up to post composition

by PSLL2 IR e conformal auto mor

by2 H Za switch FIT and it

Think Goldberg 99For a green set czif y.EE of

critical points of index 2there are at most a Catalannumber Cn of rational funtimeof degree n 11 With these criticalpoints for Psia E

Cn t.LY S a

Assuming existence of geo multichordfor each 2 Ya multi

gines Cnthx psuz.ir 12 27

rational functions also distinct forASIC Lc PSLV a equivalence

G l of

Cor There is a ungu geodesic whichµ in Xy Hl Xi Xm

or When 2 X Zayn Xun all real7 then F Cn rational funtions with

these critical pointsMoreover these rational functions are

PSL12 equivalent to a REALlF

Multi chordal SLE well defined

Part I SLE K s

kz8a Z Ge 5 Id ng t is fixed

TonightNor mushyatART IIB

Radial SLE CD target at o

jnoradited at o

ihsA iE d

ft BM on S with diffiusitity k

eiBkt que irkBeThe maximal solution time

Kt ft C B Tiz e t

Same phase transition as chordal SLEK

Thu As k soo SLEK likeconverges

to Dl e D fb test ttlD

BYogeu r si.EE is

Ssgtf

off Sect SB ds

f gear's 4 is

Local timeof Boy on

81

Definition Learner Kafarer energyN c 9p measure on S a Toi J i

pl S XI 121 for allintervals I To D

p E N peds dt

PtIdf dt

Scp J I'M f It

i

somewhat'ising anl chain

uniform measure

I pi unifmons

of if constant

Ang Park W Sip o

Then NY endowed with the weak topology

SBI Steton E m

mtgYpDpwith rare

fusion

regulanayytutFor each ft N cos 1kt

p groughulbhomeomorphism If

Contraution principleL K energy is the LDP of Radial8LEK as k ooo

EEintm