Connectedness of the Free Uniform Spanning Forest as a function of edge weights

TL;DR

This paper investigates how assigning weights to edges in a graph affects the connectivity of the free uniform spanning forest, revealing phase transitions and explicit distributions in specific cases.

Contribution

It introduces a weighted model for the FSF on Cartesian product graphs and proves phase transition phenomena and explicit formulas for particular configurations.

Findings

Weighted edges induce a phase transition in FSF connectivity

For large weights, FSF is connected; for small weights, it is disconnected

Explicit distribution formula for FSF when H is a single edge

Abstract

Let $G$ be the Cartesian product of a regular tree $T$ and a finite connected transitive graph $H$. It is shown in arXiv:2006.06387 that the Free Uniform Spanning Forest ($\mathsf{FSF}$) of this graph may not be connected, but the dependence of this connectedness on $H$ remains somewhat mysterious. We study the case when a positive weight $w$ is put on the edges of the $H$-copies in $G$, and conjecture that the connectedness of the $\mathsf{FSF}$ exhibits a phase transition. For large enough $w$ we show that the $\mathsf{FSF}$ is connected, while for a large family of $H$ and $T$, the $\mathsf{FSF}$ is disconnected when $w$ is small (relying on arXiv:2006.06387). Finally, we prove that when $H$ is the graph of one edge, then for any $w$, the $\mathsf{FSF}$ is a single tree, and we give an explicit formula for the distribution of the distance between two points within the tree.