Indistinguishability of trees in uniform spanning forests

Tom Hutchcroft*, Asaf Nachmias

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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 et al. (Ann Probab 29(1):1–65, 2001). We also answer positively two additional questions of Benjamini et al. (Ann Probab 29(1):1–65, 2001) 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.

Original languageEnglish
Pages (from-to)113-152
Number of pages40
JournalProbability Theory and Related Fields
Issue number1-2
StatePublished - 1 Jun 2017


FundersFunder number
Horizon 2020 Framework Programme676970
Natural Sciences and Engineering Research Council of Canada


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