## Annals of Probability

### Indistinguishability of the components of random spanning forests

Ádám Timár

#### 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.

#### Article information

Source
Ann. Probab., Volume 46, Number 4 (2018), 2221-2242.

Dates
Revised: August 2017
First available in Project Euclid: 13 June 2018

https://projecteuclid.org/euclid.aop/1528876826

Digital Object Identifier
doi:10.1214/17-AOP1225

Mathematical Reviews number (MathSciNet)
MR3813990

Zentralblatt MATH identifier
06919023

#### Citation

Timár, Ádám. Indistinguishability of the components of random spanning forests. Ann. Probab. 46 (2018), no. 4, 2221--2242. doi:10.1214/17-AOP1225. https://projecteuclid.org/euclid.aop/1528876826

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