Indistinguishability of trees in uniform spanning forests

@article{Hutchcroft2015IndistinguishabilityOT,
  title={Indistinguishability of trees in uniform spanning forests},
  author={Tom Hutchcroft and Asaf Nachmias},
  journal={Probability Theory and Related Fields},
  year={2015},
  volume={168},
  pages={113-152}
}
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… 
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