The Component Graph of the Uniform Spanning Forest: Transitions in Dimensions $9,10,11,\ldots$

TL;DR

This paper investigates how the connectivity properties of the component graph derived from the uniform spanning forest differ across dimensions, revealing dimension-dependent structural changes and introducing the concept of ubiquitous subgraphs.

Contribution

It introduces the notion of ubiquitous subgraphs in the component graph and hypergraph, showing how their structure varies with dimension, refining previous results on diameter growth.

Findings

Ubiquitous subgraphs change with dimension above 8

Ubiquitous subhypergraphs distinguish dimensions 5 to 8

Diameter of the component graph increases with dimension

Abstract

We prove that the uniform spanning forests of $\mathbb{Z}^d$ and $\mathbb{Z}^{\ell}$ have qualitatively different connectivity properties whenever $\ell >d \geq 4$. In particular, we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex, which we call the component graph. We introduce the notion of ubiquitous subgraphs and show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and is above $8$. To separate dimensions $5,6,7,$ and $8$, we prove a similar result concerning ubiquitous subhypergraphs in the component hypergraph. Our result sharpens a theorem of Benjamini, Kesten, Peres, and Schramm, who proved that the diameter of the component graph increases by one every time the dimension increases by four.