Random walks on the two dimensional uniform spanning tree

We study random walks on the uniform spanning tree (UST) on Z². 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 Z² is 16/13 almost surely.

In order to prove these results, we use the work of Barlow, Jarai, Kumagai, Misumi and Slade on random walks on random graphs which implies that it suffices to establish volume and effective resistance bounds for the UST. Using Wilson’s algorithm, we show that this reduces to obtaining estimates on the number of steps of loop-erased random walks (LERW) in subsets of Z². If we let M_n be the number of steps of a LERW from the origin to the circle of radius n, then Kenyon showed that E[M_n] is logarithmically asymptotic to n^5/4. In addition to this fact, we need to show that with high probability, M_n is close to its mean. In fact, we will obtain exponential moment bounds for M_n which implies that the tails of M_n decay exponentialy.

Joint work with Martin Barlow