Indistinguishability of Trees in Uniform Spanning Forests

TL;DR

This paper proves that in uniform spanning forests of unimodular random networks, the forest components are indistinguishable by invariant properties, confirming a conjecture and answering related questions about connectivity and component properties.

Contribution

It establishes the indistinguishability of forest components in unimodular networks and resolves conjectures about their connectivity and transience properties.

Findings

Components are indistinguishable by invariant properties.

FUSF is either connected or has infinitely many components.

If FUSF and WUSF differ, components are transient and infinitely-ended.

Abstract

We prove that in both the free and the wired uniform spanning forest (FUSF and WUSF) of any unimodular random rooted network (in particular, of any Cayley graph), it is impossible to distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely. This confirms a conjecture of Benjamini, Lyons, Peres and Schramm. We use this to answer positively two additional questions of Benjamini, Lyons, Peres and Schramm under the assumption of unimodularity. We prove that on any unimodular random rooted network, the FUSF is either connected or has infinitely many connected components almost surely, and, if the FUSF and WUSF are distinct, then every component of the FUSF is transient and infinitely-ended almost surely. All of these results are new even for Cayley graphs.