Infinite collision property for the three-dimensional uniform spanning tree

TL;DR

This paper provides a quantitative estimate on the number of collisions between two independent simple random walks on the three-dimensional uniform spanning tree, reinforcing the known infinite collision property.

Contribution

It offers a new quantitative analysis and proof of the infinite collision property for the 3D uniform spanning tree, complementing previous qualitative results.

Findings

Two independent random walks collide infinitely often on the 3D uniform spanning tree.

Quantitative bounds on the number of collisions are established.

Provides an alternative proof of the infinite collision property.

Abstract

Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^3$, whose probability law is denoted by $\mathbf{P}$. For $\mathbf{P}$-a.s. realization of $\mathcal{U}$, the recurrence of the the simple random walk on $\mathcal{U}$ is proved in [5] and it is also demonstrated in [8] that two independent simple random walks on $\mathcal{U}$ collide infinitely often. In this article, we will give a quantitative estimate on the number of collisions of two independent simple random walks on $\mathcal{U}$, which provides another proof of the infinite collision property of $\mathcal{U}$.