Wired cycle-breaking dynamics for uniform spanning forests
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 [Ann. Probab. 29 (2001) 1–65]. 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.
Additional Information© 2016 Institute of Mathematical Statistics. Received April 2015; revised September 2015. We thank Tel Aviv University for its hospitality while this work was completed.We also thank Omer Angel and Asaf Nachmias for many valuable comments on earlier versions of the paper, and thank the referee and Associate Editor for their comments and suggestions.
Published - 15-AOP1063.pdf
Submitted - 1504.03928.pdf