# Induced graphs of uniform spanning forests

## TL;DR

This paper studies the properties of induced graphs of components in uniform spanning forests, revealing recurrence, resistance growth, and resampling properties in various graphs, especially in high dimensions.

## Contribution

It establishes new recurrence and resistance growth results for induced graphs of WSF components and introduces a resampling property for vertex-transitive graphs.

## Key findings

- Induced graphs of WSF components are recurrent in dimensions ≥8.
- Effective resistance between points grows linearly in dimensions ≥9.
- Resampling property links WSF components conditioned on their induced graphs.

## Abstract

Given a subgraph $H$ of a graph $G$, the induced graph of $H$ is the largest subgraph of $G$ whose vertex set is the same as that of $H$. Our paper concerns the induced graphs of the components of $\operatorname{WSF}(G)$, the wired spanning forest on $G$, and, to a lesser extent, $\operatorname{FSF}(G)$, the free uniform spanning forest. We show that the induced graph of each component of $\operatorname{WSF}(\mathbb Z^d$) is almost surely recurrent when $d\ge 8$. Moreover, the effective resistance between two points on the ray of the tree to infinity within a component grows linearly when $d\ge9$. For any vertex-transitive graph $G$, we establish the following resampling property: Given a vertex $o$ in $G$, let $\mathcal T_o$ be the component of $\operatorname{WSF}(G)$ containing $o$ and $\overline{\mathcal{T}_o}$ be its induced graph. Conditioned on $\overline{\mathcal{T}_o}$, the tree $\mathcal T_o$ is distributed as $\operatorname{WSF}(\overline{\mathcal{T}_o})$. For any graph $G$, we also show that if $\mathcal T_o$ is the component of $\operatorname{FSF}(G)$ containing $o$ and $\overline{\mathcal{T}_o}$ is its induced graph, then conditioned on $\overline{\mathcal{T}_o}$, the tree $\mathcal T_o$ is distributed as $\operatorname{FSF}(\overline{\mathcal{T}_o})$.