Wired Cycle-Breaking Dynamics for Uniform Spanning Forests

TL;DR

This paper proves that in certain random graphs, the components of the wired uniform spanning forest are almost surely one-ended, extending previous results by removing degree restrictions and introducing new Markov chain techniques.

Contribution

It establishes the one-endedness of WUSF components in all transient reversible graphs without degree bounds, using wired cycle-breaking dynamics.

Findings

WUSF components are one-ended in all transient reversible graphs.

Confirmed the conjecture for supercritical Galton-Watson trees.

Introduced wired cycle-breaking dynamics as a new analytical tool.

Abstract

We prove that every component of the wired uniform spanning forest (WUSF) is one-ended almost surely in every transient reversible random graph, removing the bounded degree hypothesis required by earlier results. We deduce that every component of the WUSF is one-ended almost surely in every supercritical Galton-Watson tree, answering a question of Benjamini, Lyons, Peres and Schramm. Our proof introduces and exploits a family of Markov chains under which the oriented WUSF is stationary, which we call the wired cycle-breaking dynamics.