Geometry of the uniform spanning forest components in high dimensions

TL;DR

This paper investigates the geometric structure and connectivity properties of the uniform spanning forest components in high-dimensional integer lattices, providing foundational estimates relevant to related probabilistic models.

Contribution

It offers new insights into the geometry and connectivity of uniform spanning forest components in dimensions five and higher, supplementing existing knowledge with preliminary estimates.

Findings

Connectivity properties with respect to Euclidean distance

Connectivity properties with respect to intrinsic distance

Preliminary estimates used in sandpile model research

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.