cover times, blanket times, and the gff
DESCRIPTION
Jian Ding. Berkeley-Stanford-Chicago. cover times, blanket times, and the GFF. James R. Lee. University of Washington. Yuval Peres. Microsoft Research. random walks on graphs. By putting conductances { c uv } on the edges of the graph, we can get any reversible Markov chain. - PowerPoint PPT PresentationTRANSCRIPT
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cover times, blanket times, and the GFF
Jian DingBerkeley-Stanford-Chicago
James R. LeeUniversity of Washington
Yuval PeresMicrosoft Research
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random walks on graphs
By putting conductances {cuv} on the edges of the graph,we can get any reversible Markov chain.
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hitting and coveringHitting time: H(u,v) = expected # of steps to hit v starting at u
Commute time: κ(u,v) = H(u,v) + H(v,u)(metric)
Cover time: tcov(G) = expected time to hit all vertices of G, starting from the worst vertex
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hitting and covering
pathcomplete graph
expander2-dimensional grid3-dimensional grid
complete d-ary tree
n2
n log nn log nn (log n)2
n log nn (log n)2/log d
orders of magnitude of some cover times
[coupon collecting][Broder-Karlin 88][Aldous 89, Zuckerman 90][Aldous 89, Zuckerman 90][Zuckerman 90]
Hitting time: H(u,v) = expected # of steps to hit v starting at uCommute time: κ(u,v) = H(u,v) + H(v,u)
(metric)Cover time: tcov(G) = expected time to hit all vertices of G, starting from the worst vertex
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hitting and covering
(1-o(1)) n log n · tcov(G) · 2nm
general bounds (n = # vertices, m = # edges)
[Alelinuas-Karp-Lipton-Lovasz-Rackoff’79][Feige’95, Matthews’88]
Hitting time: H(u,v) = expected # of steps to hit v starting at uCommute time: κ(u,v) = H(u,v) + H(v,u)
(metric)Cover time: tcov(G) = expected time to hit all vertices of G, starting from the worst vertex
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electrical resistance
+u v
Reff (u,v) = inverse of electrical current flowing from u to v
[Chandra-Raghavan-Ruzzo-Smolensky-Tiwari’89]:If G has m edges, then for every pair u,v
κ(u,v) = 2m Reff (u,v)
(endows κ with certaingeometric properties)
Hitting time: H(u,v) = expected # of steps to hit v starting at uCommute time: κ(u,v) = H(u,v) + H(v,u)
(metric)Cover time: tcov(G) = expected time to hit all vertices of G, starting from the worst vertex
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computationHitting time: Easy to compute in deterministic poly time by solving system of linear equationsH(u,u) = 0 H(u,v) = 1 + w»u H(w,v)
Cover time: Easy to compute in exponential time by a determinsitic algorithm and polynomial time by a randomized algorithm
Natural question: Does there exist a poly-time deterministically computable O(1)-approximation for general graphs?
[Aldous-Fill’94]
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approximation in derministic poly-time
Clearly:
[Kahn-Kim-Lovasz-Vu’99] show that Matthews’ lower bound gives an O(log log n)2 approximation to tcov
[Matthews’88] proved:
Hmax = maxu;v2V
H(u;v) · tcov
tcov · Hmax(1+ logn)and tcov ¸ max
Sµ Vmin
u;v2SH (u;v)(logjSj ¡ 1)
[Feige-Zeitouni’09] give a (1+²)-approximation for trees for every ² > 0, using recursion.
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blanket times
Blanket times [Winkler-Zuckerman’96]:The ¯-blanket time tblanket(G,¯) is the expected first time T at which all the local times,
are within a factor of ¯.
LxT = # visits to x
¼(x)
100 5 3
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blanket times
Blanket times [Winkler-Zuckerman’96]:The ¯-blanket time tblanket(G,¯) is the expected first time T at which all the local times,
are within a factor of ¯.
LxT = # visits to x
¼(x)
0.01 0.2 0.3330.99
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blanket time conjecture
Conjecture [Winkler-Zuckerman’96]: For every graph G and 0 < ¯ < 1, tblanket(G,¯) ³ tcov(G).
Proved for some special cases.
³ ³
True up to (log log n)2 by [Kahn-Kim-Lovasz-Vu’99]
³
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Gaussian free field on a graphA centered Gaussian process {gv}v2V satisfying for all u,v 2 V:
E (gu ¡ gv)2 = Re®(u;v)and for some fixed v0 2 V.gv0 = 0
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Gaussian free field on a graphA centered Gaussian process {gv}v2V satisfying for all u,v 2 V:
E (gu ¡ gv)2 = Re®(u;v)and for some fixed v0 2 V.gv0 = 0
Density proportional to
e¡ 12
Pu» v jgu¡ gvj2
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Gaussian free field on a graphA centered Gaussian process {gv}v2V satisfying for all u,v 2 V:
E (gu ¡ gv)2 = Re®(u;v)and for some fixed v0 2 V.gv0 = 0
Covariance given by the Green function of SRW killed at v0 :
expected # visits to v from RW start at u before hitting v0
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main result
For every graph G=(V, E),
tcov(G) ³ jE jµ
E maxv2V
gv¶2
³ ¯ tblanket(G;¯ )
for every 0 < ¯ < 1, where {gv}v2V is the GFF on G.
Positively resolves the Winkler-Zuckerman blanket time conjectures.
³ ³ ³
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geometrical interpretation embedded in Euclidean space via GFF.
where D is the diameter of the projection of this embeddingonto a random line:
tcov(G) ³ n ¢jE j(ED)2
𝑫An illustration for the complete graph
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Gaussian processes
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Gaussian processes
Consider a Gaussian process {Xu : u 2 S} with E (Xu)=0 8u2S
Such a process comes with a natural metricd(u;v) =
qE (X u ¡ X v)2
transforming (S, d) into a metric space.
PROBLEM: What is E max { Xu : u 2 S } ?
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majorizing measures
Majorizing measures theorem [Fernique-Talagrand]:
E supu2S
X u ³ °2(S;d)where d(u;v) =
qE (X u ¡ X v)2.
is a functional on metric spaces, given by an explicit formula which takes exponential time to calculate from the definition
°2
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main theorem, restated
For every graph G=(V, E),
tcov(G) ³³°2(V;p · )
´2
where · = commute time.
We also construct a deterministic poly-time algorithm to approximate within a constant factor.°2
THEOREM:
COROLLARY: There is a deterministic poly-time O(1)-approximation for tcov
Positively answers the question of Aldous and Fill.
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majorizing measures
Majorizing measures theorem [Fernique-Talagrand]:
E supu2S
X u ³ °2(S;d)where d(u;v) =
qE (X u ¡ X v)2.
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Gaussian processes
PROBLEM: What is E max { Xu : u 2 S } ?
®If random variables are “independent,”expect the union bound to be tight.
Gaussian concentration:
Expect max for k points is about
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Gaussian processesGaussian concentration:
Sudakov minoration:
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chaining
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chaining
Best possible tree upper bound yields
°2
E sups2S
X s .Z 10
qlog N (S;d; ") d"[Dudley’67]:
where N(S,d,²) is the minimal number of ²-balls needed to cover S
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hints of a connectionGaussian concentration:
Sudakov minoration:
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hints of a connection
Sudakov minoration:
Matthew’s bound (1988):
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hints of a connectionGaussian concentration:
KKLV’99 concentration:
global time.is local time at and
ℙ (𝐿𝑇𝑢 −𝐿𝑇
𝑣>𝛼 )≤ exp (− 𝛼2
4 ℓ 𝑅eff (𝑢 ,𝑣 ) )For all nodes and
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hints of a connectionDudley’67 entropy bound:
Barlow-Ding-Nachmias-Peres’09 analog for cover times
E sups2S
X s .Z 10
qlog N (S;d;") d"
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hints of a connection
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Dynkin isomorphism theory
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Dynkin isomorphism theoryRecall the local time of v at time t is given by
L vt = # visits to vdeg(v) :
Ray-Knight (1960s):Characterize the local times of Brownian motion.Dynkin (1980):General connection of Markov process to Gaussian fields.The version we used is due toEisenbaum, Kaspi, Marcus, Rosen, and Shi (2000)
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Generalized Ray-Knight theoremFor define .Let be the GFF on with Then
where
law
𝑣0𝑣
¿law
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blanket time vs. GFF
𝐿𝑇𝑥+12 𝑔𝑥
2=−⋅𝑔𝑥+12𝑔𝑥
2ℓ𝑙𝑎𝑤ℓ−
√2 ℓ 𝑔𝑥−12 𝑔𝑥
2
𝑉
√2 ℓ? 𝐿𝑇
𝑥
+¿12𝑔𝑥
2
𝑉
¿
𝑇=𝑇 (ℓ)M = maxx gx ` À (EM )2
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the lower bound ³ |E|
law
To lower bound cover time: Need to show that for with good chance
small small w.h.p.
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only one Everest, but...
𝜖𝔼𝑀
𝔼𝑀
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a problem on Gaussian processes
Gaussian free field: 𝑀=max
𝑥∈𝑉𝑔𝑥
𝜖𝔼𝑀 𝔼𝑀
{𝑔𝑥 }𝑥∈𝑉
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a problem on Gaussian processes
Gaussian free field: 𝑀=max
𝑥∈𝑉𝑔𝑥
{𝑔𝑥 }𝑥∈𝑉
We need strong estimates on the size of this window.(want to get a point there with probability at least 0.1)Problem: Majorizing measures handles first moments, but we need second moment bounds.
𝜖𝔼𝑀 𝔼𝑀
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percolation on trees and the GFF
𝔼𝑀
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percolation on trees and the GFF
𝜖𝔼𝑀 𝔼𝑀
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percolation on trees and the GFF
First and second moments agree for
percolation on balanced trees
Problem: General Gaussian processes behaves nothing like percolation!Resolution: Processes coming from the
Isomorphism Theorem all arise from a GFF
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percolation on trees and the GFF
First and second moments agree for
percolation on balanced trees
For DGFFs, using electrical network theory, we show that it is possibleto select a subtree of the MM tree and a delicate filtration of theprobability space so that the Gaussian process can be coupled to apercolation process.
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example: precise asymptotics in 2D
Bolthausen, Deuschel and Giacomin (2001):
Dembo, Peres, Rosen, and Zeitouni (2004): For 2D torus𝑡 cov∼|𝐸|¿
For Gaussian free field on lattice of vertices
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open questions
Lower bound holds for: - complete graph - complete d-ary tree - discrete 2D torus - bounded degree graphs [Ding’11]QUESTION: Is there a deterministic, polynomial-time
(1+²)-approximation to the cover time for every ² > 0 ?QUESTION: Is the standard deviation of the time-to-cover bounded by the maximum hitting time?
𝑡 cov∼|𝐸|¿Upper bound is true.