Scaling limits of the three-dimensional uniform spanning tree and associated random walk

Abstract

We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant laws actually converge along a particular scaling sequence. The techniques that we use to establish these results are further applied to obtain various properties of the intrinsic metric and measure of any limiting space, including showing that the Hausdorff dimension of such is given by $3/\mathit{\beta }$, where $\mathit{\beta }\approx 1.624\dots$ is the growth exponent of three-dimensional loop-erased random walk. Additionally, we study the random walk on the three-dimensional uniform spanning tree, deriving its walk dimension (with respect to both the intrinsic and Euclidean metric) and its spectral dimension, demonstrating the tightness of its annealed law under rescaling, and deducing heat kernel estimates for any diffusion that arises as a scaling limit.