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July 2018 Indistinguishability of the components of random spanning forests
Ádám Timár
Ann. Probab. 46(4): 2221-2242 (July 2018). DOI: 10.1214/17-AOP1225

Abstract

We prove that the infinite components of the Free Uniform Spanning Forest (FUSF) of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the Free Minimal Spanning Forest (FMSF). We also show that with the above assumptions there can only be 0, 1 or infinitely many components, which solves the problem for the FUSF of Caylay graphs completely. These answer questions by Benjamini, Lyons, Peres and Schramm for Cayley graphs, which have been open up to now. Our methods apply to a more general class of percolations, those satisfying “weak insertion tolerance”, and work beyond Cayley graphs, in the more general setting of unimodular random graphs.

Citation

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Ádám Timár. "Indistinguishability of the components of random spanning forests." Ann. Probab. 46 (4) 2221 - 2242, July 2018. https://doi.org/10.1214/17-AOP1225

Information

Received: 1 January 2017; Revised: 1 August 2017; Published: July 2018
First available in Project Euclid: 13 June 2018

zbMATH: 06919023
MathSciNet: MR3813990
Digital Object Identifier: 10.1214/17-AOP1225

Subjects:
Primary: 60D05
Secondary: 82B43

Keywords: Indistinguishability , insertion tolerance , Minimal spanning forest , Spanning forests , Uniform spanning forest

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 4 • July 2018
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