The number of ends in the uniform spanning tree for recurrent unimodular random graphs

TL;DR

This paper proves that for recurrent unimodular random graphs, the uniform spanning tree's number of ends matches the graph's ends, resolving a longstanding conjecture and completing the understanding for all cases.

Contribution

It establishes the almost sure equality of the number of ends between the graph and its uniform spanning tree in the recurrent case, confirming a conjecture by Aldous and Lyons.

Findings

Number of ends of the uniform spanning tree equals the graph's ends in recurrent unimodular graphs

Complete resolution of the ends problem for wired uniform spanning forests in unimodular graphs

Confirms the Aldous-Lyons conjecture from 2006

Abstract

We prove that if a unimodular random rooted graph is recurrent, the number of ends of its uniform spanning tree is almost surely equal to the number of ends of the graph. Together with previous results in the transient case, this completely resolves the problem of the number of ends of wired uniform spanning forest components in unimodular random rooted graphs and confirms a conjecture of Aldous and Lyons (2006).