Infinite collision property for the three-dimensional uniform spanning tree

Abstract

Let [Formula: see text] be the three-dimensional uniform spanning tree, whose probability law is denoted by [Formula: see text]. For [Formula: see text]-a.s. realization of [Formula: see text], the recurrence of the simple random walk on [Formula: see text] is proved in Benjamini et al. (2001) [I. Benjamini, R. Lyons, Y. Peres and O. Schramm, Uniform spanning forests, Ann. Probab. 29(1) (2001) 1–65] and it is also demonstrated in Hutchcroft and Peres (2015) [T. Hutchcroft and Y. Peres, Collisions of random walks in reversible random graphs, Electron. Commun. Probab. 20(63) (2015) 1–6] that two independent simple random walks on [Formula: see text] collide infinitely often. In this paper, we will give a quantitative estimate on the number of collisions of two independent simple random walks on [Formula: see text], which provides another proof of the infinite collision property of [Formula: see text].