Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree

TL;DR

This paper studies the scaling limits of the two-dimensional uniform spanning tree and the behavior of simple random walks on it, revealing fractal dimensions and diffusion properties in the limit.

Contribution

It establishes tightness of the scaled uniform spanning trees and random walks, characterizes their limit objects, and derives transition density estimates for the diffusions.

Findings

Hausdorff dimension of limits is almost surely 8/5

Tightness of the scaled trees and walks is proved

Transition densities for the limiting diffusions are obtained

Abstract

The first main result of this paper is that the law of the (rescaled) two-dimensional uniform spanning tree is tight in a space whose elements are measured, rooted real trees continuously embedded into Euclidean space. Various properties of the intrinsic metrics, measures and embeddings of the subsequential limits in this space are obtained, with it being proved in particular that the Hausdorff dimension of any limit in its intrinsic metric is almost surely equal to $8/5$. In addition, the tightness result is applied to deduce that the annealed law of the simple random walk on the two-dimensional uniform spanning tree is tight under a suitable rescaling. For the limiting processes, which are diffusions on random real trees embedded into Euclidean space, detailed transition density estimates are derived.