Spectral Dimension and Random Walks on the Two Dimensional Uniform Spanning Tree

Abstract

We study the simple random walk on the uniform spanning tree on $${\mathbb {Z}^2}$$ . 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 $${\mathbb {Z}^2}$$ is 16/13 almost surely.