TY - JOUR

T1 - Indistinguishability of trees in uniform spanning forests

AU - Hutchcroft, Tom

AU - Nachmias, Asaf

N1 - Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.

PY - 2017/6/1

Y1 - 2017/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84963724148&partnerID=8YFLogxK

U2 - 10.1007/s00440-016-0707-3

DO - 10.1007/s00440-016-0707-3

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AN - SCOPUS:84963724148

SN - 0178-8051

VL - 168

SP - 113

EP - 152

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 1-2

ER -