Wired Cycle-Breaking Dynamics for Uniform Spanning Forests

Authors :: Tom Hutchcroft
Hutchcroft, Thomas [0000-0003-0061-593X]
Apollo - University of Cambridge Repository
Source :: Ann. Probab. 44, no. 6 (2016), 3879-3892
Publication Year :: 2015
Publisher :: arXiv, 2015.
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.<br />Comment: 15 pages, 1 figure