Indistinguishability of Trees in Uniform Spanning Forests
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 infinitelyended almost surely. All of these results are new even for Cayley graphs.
 Publication:

arXiv eprints
 Pub Date:
 June 2015
 arXiv:
 arXiv:1506.00556
 Bibcode:
 2015arXiv150600556H
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 43 pages, 2 figures. Version 2: minor corrections and improvements