Triple collisions on a comb graph

TL;DR

This paper investigates the collision behavior of three independent random walks on a specially constructed comb graph, revealing a phase transition in collision frequency depending on a parameter controlling the vertical segment length.

Contribution

It extends previous work on two random walks to three, identifying a phase transition in collision frequency based on the vertical segment truncation parameter.

Findings

For α ≤ 1, three random walks collide infinitely often almost surely.

For α > 1, they collide only finitely often almost surely.

The results generalize known phase transitions from two to three random walks.

Abstract

In this article, we consider the number of collisions of three independent simple random walks on a subgraph of the two-dimensional square lattice obtained by removing all horizontal edges with vertical coordinate not equal to 0 and then, for $n\in \mathbb{Z}$, restricting the vertical segment of the graph located at horizontal coordinate $n$ to the interval $\{0,1,\dots,\log^{\alpha}(|n|\vee 1)\}$. Specifically, we show the following phase transition: when $\alpha\leq 1$, the three random walks collide infinitely many times almost-surely, whereas when $\alpha>1$, they collide only finitely many times almost-surely. This is a variation of a result of Barlow, Peres and Sousi, who showed a similar phase transition for two random walks when the vertical segments are truncated at height $|n|^{\alpha}$.