Random walk on the infinite cluster of the percolation model

Grimmett, Geoffrey; Kesten, Harry; Zhang, Y.

doi:10.1007/bf01195881

Cited by 82 publications

(81 citation statements)

References 11 publications

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“…Grimmett, Kesten and Zhang [20] proved via "electrostatic techniques" that this random walk is transient in d ≥ 3; extensions concerning the existence of various "energy flows" appeared in [1,24,26,29,33]. A great amount of effort has been spent on deriving estimates on the heat-kernel-i.e., the probability that the walk is at a particular site Fig.…”

Section: Discussion and Related Workmentioning

confidence: 99%

Quenched invariance principle for simple random walk on percolation clusters

Berger

Biskup²

2006

Probab. Theory Relat. Fields

193

258

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Abstract. We consider the simple random walk on the (unique) infinite cluster of supercritical bond percolation in Z d with d ≥ 2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.

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Section: Discussion and Related Workmentioning

confidence: 99%

Quenched invariance principle for simple random walk on percolation clusters

Berger

Biskup²

2006

Probab. Theory Relat. Fields

193

258

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“…We say the G is transient if the random walk is transient, and we call G recurrent otherwise. Initiated by the results of paper [16], several authors have considered the question of whether or not an infinite open graph generated by a three-dimensional percolation model is almost surely transient. Results for undirected percolation include [16,18,20], and the directed case has been studied in [5] using the method of 'unpredictable paths'.…”

Section: Principal Resultsmentioning

confidence: 99%

“…A similar result was proved in [5] for directed percolation in three dimensions, whenever the edge density is sufficiently large. The methods used in the latter paper are quite different from those used in [16], and have other applications also. We present in Theorem 3 a positive answer to a question posed in [5], namely whether the transience result for directed percolation may be extended to all values of p satisfying p > p c .…”

Section: Introductionmentioning

confidence: 99%

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Directed Percolation and Random Walk

Grimmett

Hiemer

2002

In and Out of Equilibrium

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Abstract. Techniques of 'dynamic renormalization', developed earlier for undirected percolation and the contact model, are adapted to the setting of directed percolation, thereby obtaining solutions of several problems for directed percolation on Z d where d ≥ 2. The first new result is a type of uniqueness theorem: for every pair x and y of vertices which lie in infinite open paths, there exists almost surely a third vertex z which is joined to infinity and which is attainable from x and y along directed open paths. Secondly, it is proved that a random walk on an infinite directed cluster is transient, almost surely, when d ≥ 3. And finally, the block arguments of the paper may be adapted to systems with infinite range, subject to certain conditions on the edge probabilities.

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“…As a corollary we conclude that the (a.s. unique) percolation cluster is a.s. transient when p > p c . This follows from the transience of the percolation cluster on Z 3 (see [13,14]). Note also that for Gupta-Sidki groups a similar construction of lifted subgroups gives immediately Z 3 + .…”

Section: Odds and Endsmentioning

confidence: 99%

Percolation on Grigorchuk Groups

Muchnik¹,

Pak²

2001

Communications in Algebra

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Let p c (G) be the critical probability of the site percolation on the Cayley graph of group G. In [2] of Benjamini and Schramm conjectured thatp c < 1, given the group is infinite and not a finite extension of Z. The conjecture was proved earlier for groups of polynomial and exponential growth and remains open for groups of intermediate growth.In this note we prove the conjecture for a special class of Grigorchuk groups, which is a special class of groups of intermediate growth. The proof is based on an algebraic construction. No previous knowledge of percolation is assumed.

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Random walk on the infinite cluster of the percolation model

Cited by 82 publications

References 11 publications

Quenched invariance principle for simple random walk on percolation clusters

Quenched invariance principle for simple random walk on percolation clusters

Directed Percolation and Random Walk

Percolation on Grigorchuk Groups

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