Collisions of random walks on comb graphs with a planar base

TL;DR

This paper investigates how the number of collisions between two independent random walks on comb graphs varies with the growth of the teeth, revealing phase transitions and answering a previously open question.

Contribution

It establishes a phase transition in collision behavior on comb graphs with planar bases, including a specific case with logarithmic growth, and addresses an open problem.

Findings

Finitely many collisions occur if the growth parameter exceeds 1.

Infinitely many collisions occur if the growth parameter is at most 1.

Phase transitions depend on the structure of the base graph and the growth of teeth.

Abstract

In this article we study collisions of two independent random walks on comb graphs $\mathrm{Comb}(\tilde{G},f)$ for a large class of recurrent planar graphs $\tilde{G}$ and profile functions $f$, the latter governing the length of vertical segments (called "teeth") attached to vertices of the base graph $\tilde{G}$. We prove that the number of collisions of two random walks starting from the same site undergoes a phase transition depending on the growth of $f$. As a benchmark example, we show that for $\mathrm{Comb}(\mathbb{Z}^2,f_\gamma)$ with $f_\gamma(z) = \log^{\gamma}(\|z\|_\infty\vee 1)$ and $\|\cdot \|_\infty$ denoting the supremum norm, two independent random walks started at the origin collide finitely often almost surely if $\gamma > 1$, answering a question of Barlow, Peres, and Sousi, see arXiv:1003.3255, who established that infinitely many collisions occur almost surely if $\gamma \leq 1$. We furthermore establish phase transitions in the cases where the base graph $\tilde{G}$ is pre-fractal, or a typical realization of a supercritical cluster of planar Bernoulli bond percolation.