Indistinguishability of trees in uniform spanning forests
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Publication:2359739
DOI10.1007/S00440-016-0707-3zbMATH Open1407.60019arXiv1506.00556OpenAlexW565170798MaRDI QIDQ2359739
Author name not available (Why is that?)
Publication date: 22 June 2017
Published in: (Search for Journal in Brave)
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 infinitely-ended almost surely. All of these results are new even for Cayley graphs.
Full work available at URL: https://arxiv.org/abs/1506.00556
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