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

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.