Quenched invariance principles for walks on clusters of percolation or among random conductances

Abstract

In this work we principally study random walk on the supercritical infinite cluster for bond percolation on Z d . We prove a quenched functional central limit theorem for the walk when d ≥ 4. We also prove a similar result for random walk among i.i.d. random conductances along nearest neighbor edges of Z d , when d ≥ 1. IntroductionConsider supercritical bond-percolation on Z d , d ≥ 2, and the simple random walk on the infinite cluster, which at each jump picks with equal probability one of the neighboring sites in the infinite cluster. Is it true that for a.e. configuration such that the origin belongs to the infinite cluster, the random walk starting at the origin exits a symmetric slab through either side with probability tending to 1 2 , as the width of the slab tends to infinity?We give in this work a positive answer to this question when d ≥ 4. This answer comes as a consequence of a more general quenched invariance principle for the walk on the supercritical infinite cluster, when d ≥ 4. We also discuss the related case of a walk evolving in a network of i.i.d. random conductances placed along nearest-neighbor edges of Z d . We are able to prove a quenched invariance principle in this situation, for general d ≥ 1. For partial results in this direction, we refer to Anshelevich-Khanin-Sinai [2], Boivin [5], Boivin-Depauw [6], Kozlov [19].Before discussing our results any further, we describe the models more precisely. We begin with random walk on the supercritical infinite cluster. We let B d stand for the set of nearest-neighbor bonds (or edges) on Z d , d ≥ 2, and = {0, 1} B d for the set of configurations. We denote by P the product measure on endowed with its canonical σ -algebra A, under which the canonical coordinates, ω(b), b ∈ B d , are Bernoulli variables with success probability p ∈ (0, 1). V. Sidoravicius: IMPA, Estrada Dona Castorina 110, Jardim Botanico, CEP 22460-320, Rio de Janeiro, RJ, Brasil. e-mail: vladas@impa.br V. Sidoravicius would like to thank the FIM for financial support and hospitality during his multiple visits to ETH. His research was also partially supported by FAPERJ and CNPq. A.-S. Sznitman:

Quenched invariance principle for simple random walk on percolation clusters

“…Note: At the time a preprint version of this paper was first circulated, we learned that Mathieu and Piatnitski had announced a proof of the same result (albeit in continuous-time setting). Their proof, which has in the meantime been posted [30], is close in spirit to that of Theorem 1.1 of [38]; the main tools are Poincaré inequalities, heat-kernel estimates and homogenization theory.…”

“…(The results of [13,14] were primarily two-dimensional but, with the help of [3], they apply to all d ≥ 2; cf [38].) A number of proofs of quenched invariance principles have appeared in recent years for the cases where an annealed principle was already known.…”

“…A number of proofs of quenched invariance principles have appeared in recent years for the cases where an annealed principle was already known. The most relevant paper is that of Sidoravicius and Sznitman [38] which established Theorem 1.2 for random walk among random conductances in all d ≥ 1 and, using a very different method, also for random walk on percolation in d ≥ 4. (Thus our main theorem is new only in d = 2, 3.)…”

“…In [38] Sidoravicius and Sznitman asked the following question: Is it true that for a.e. configuration in which the origin belongs to the infinite cluster, the random walk started at the origin exits the infinite symmetric slab {(x 1 , .…”

“…In this paper we extend the desired conclusion to all d ≥ 2. As in [38], we will do so by proving a quenched invariance principle for the paths of the walk.…”