The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an Overview

Dayue Chen,Yuval Peres

doi:10.46298/dmtcs.3340

Abstract

Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph $G$, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain amenable groups known as "lamplighter groups''.We prove that zero speed for random walk on a lamplighter group implies zero speed for random walk on an infinite cluster, for any supercritical percolation parameter $p$. For $p$ large enough, we also establish the converse. We prove that if $G$ has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter $p$ sufficiently close to 1; We also show that positivity of the anchored expansion constant is preserved under a random stretch if, and only if, the stretching law has an exponential tail.

Highlights

Denote by V and E, res¡ pectively, the se¡ ts of vertices and edges of an infinite graph
We usually identify the percolation ω with the subgraph of consisting of all open edges and their end-vertices
Kesten and Zhang (1993) showed that simple random walk on the infinite cluster of supercritical Bernoulli percolation in d is transient for d 3; in other words, in Euclidean lattices, transience is preserved when the whole lattice is replaced by the infinite percolation cluster

Summary

Introduction

Denote by V and E , res¡ pectively, the se¡ ts of vertices and edges of an infinite graph. Benjamini, Lyons and Schramm (1999)¡ , abbreviated as BLS (1999) hereafter, initiated a systematic study of the¡ properties of a transitive graph that are preserved under random perturbations such as passing from to an infinite. Dayue Chen and Yuval Peres percolation cluster They conjectured that positivity of the speed limn Xn ¡ n for simple random walk Xn is preserved, where x is the graph distance from x to o. If is a Cayley graph on which simple random walk has positive speed, a.s., simple random walk on each infinite cluster of p-Bernoulli percolation has positive speed. ¡ If is a Cayley graph on which simple random walk has zero speed, a.s., simple random walk on every cluster of Bernoulli percolation has zero speed If is a Cayley graph on which simple random walk has positive speed, a.s., simple random walk on each infinite cluster of p-Bernoulli percolation has positive speed. ¡ If is a Cayley graph on which simple random walk has zero speed, a.s., simple random walk on every cluster of Bernoulli percolation has zero speed

For S S and

The lamplighter groups

Results

Theorem the graph