Wired cycle-breaking dynamics for uniform spanning forests

Abstract

We prove that every component of the wired uniform spanning forest (${\mathsf{WUSF}}$) 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 ${\mathsf{WUSF}}$ is one-ended almost surely in every supercritical Galton–Watson tree, answering a question of Benjamini, Lyons, Peres and Schramm [Ann. Probab. 29 (2001) 1–65].

Our proof introduces and exploits a family of Markov chains under which the oriented ${\mathsf{WUSF}}$ is stationary, which we call the wired cycle-breaking dynamics.