Collisions of Random Walks in Reversible Random Graphs

TL;DR

This paper proves that in certain infinite random graphs, two independent random walks starting at the same point will collide infinitely often, extending to models like uniform planar maps and critical percolation clusters.

Contribution

It establishes the almost sure infinite collisions of independent random walks in all recurrent reversible random rooted graphs, including specific models like UIPT, UIPQ, and IIC in .

Findings

Random walks collide infinitely often in recurrent reversible graphs

Applicable to UIPT, UIPQ, and IIC models in 

Extends known collision results to a broad class of random graphs

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 $\mathbb{Z}^2$.