Collisions of Random Walks

Abstract

A recurrent graph $G$ has the infinite collision property if two independent random walks on $G$, started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton-Watson tree with finite variance conditioned to survive, the incipient infinite cluster in ${\mathbb{Z}}^d$ with $d\ge 19$ and the uniform spanning tree in ${\mathbb{Z}}^2$ all have the infinite collision property. For power-law combs and spherically symmetric trees, we determine precisely the phase boundary for the infinite collision property.

31 pages, 4 figures

References in corpus (1)

• Random walks on supercritical percolation clusters