Collisions of random walks in reversible random graphs

Tom Hutchcroft,Yuval Peres

doi:10.1214/ecp.v20-4330

Abstract

We prove that in any recurrent reversible random rooted graph, two independent simple random walks started at the same vertex collide infinitely often almost surely. This applies to the Uniform Infinite Planar Triangulation and Quadrangulation and to the Incipient Infinite Cluster in $\mathbb{Z}^2$.

Highlights

Let G be an infinite, connected, locally finite graph
G is said to have the infinite collision property if for every vertex v of G, two independent random walks Xn n≥0 and Yn n≥0 started from v collide infinitely often almost surely (a.s.). Transitive recurrent graphs such as Z and Z2 are seen to have the infinite collision property, Krishnapur and Peres [14] showed that the infinite collision property does not hold for the comb graph, a subgraph of Z2
Chen and Chen [8] proved that the infinite cluster of supercritical Bernoulli bond percolation in Z2 a.s. has the infinite collision property

Summary

Introduction

Let G be an infinite, connected, locally finite graph. G is said to have the infinite collision property if for every vertex v of G, two independent random walks Xn n≥0 and Yn n≥0 started from v collide (i.e. occupy the same vertex at the same time) infinitely often almost surely (a.s.). Peres and Sousi [4] gave a sufficient condition for the infinite collision property in terms of the Green function They deduced that several classical random recurrent graphs have the infinite collision property, including the incipient infinite percolation cluster in dimensions d ≥ 19. In this note we prove that the infinite collision property holds a.s. for a large class of random recurrent graphs. Let Cn be the largest cluster of a critical Bernoulli bond percolation on the box [0, n]2 and let ρn be a uniformly random vertex of Cn. Járai [11] showed that the IIC can be defined as the weak limit of the random rooted graphs (Cn, ρn), and is unimodular [2, §2]. If G is a bipartite graph with the infinite collision property, two independent random walks on G will collide infinitely often if and only if their starting points are at an even distance from each other

Definitions

Extensions

Continuous-time random walk