Spectral dimension and random walks on the two dimensional uniform spanning tree

TL;DR

This paper investigates the behavior of simple random walks on the two-dimensional uniform spanning tree, providing estimates for transition probabilities, distances, and exit times, and establishing the spectral dimension as 16/13.

Contribution

It is the first to rigorously determine the spectral dimension of the 2D uniform spanning tree as 16/13 and to analyze random walk properties on it.

Findings

Spectral dimension of the 2D uniform spanning tree is 16/13.

Provides estimates for transition probabilities and exit times.

Analyzes the walk's distance from the starting point after n steps.

Abstract

We study simple random walk on the uniform spanning tree on Z^2 . We obtain estimates for the transition probabilities of the random walk, the distance of the walk from its starting point after n steps, and exit times of both Euclidean balls and balls in the intrinsic graph metric. In particular, we prove that the spectral dimension of the uniform spanning tree on Z^2 is 16/13 almost surely.