Geometry of the uniform spanning forest components in high dimensions

Abstract

In this note we study the geometry of the component of the origin in the Uniform Spanning Forest of $\mathbb{Z}^d$, as well as in the Uniform Spanning Tree of wired subgraphs of $\mathbb{Z}^d$, when $d \ge 5$. In particular, we study connectivity properties with respect to the Euclidean and the intrinsic distance. We intend to supplement these with further estimates in the future. We are making this preliminary note available, as one of our estimates is used in work of Bhupatiraju, Hanson and J\'arai on sandpiles.