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Indistinguishability of trees in uniform spanning forests


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.

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We are grateful to Russ Lyons for many comments, corrections and improvements to the manuscript, and also to Ander Holroyd and Yuval Peres for useful discussions. TH thanks Tel Aviv University and both authors thank the Issac Newton Institute, where part of this work was carried out, for their hospitality. This Project is supported by NSERC.

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Hutchcroft, T., Nachmias, A. Indistinguishability of trees in uniform spanning forests. Probab. Theory Relat. Fields 168, 113–152 (2017).

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Mathematics Subject Classification

  • 60D05
  • 60K99
  • 05C80