Annals of Probability
- Ann. Probab.
- Volume 46, Number 4 (2018), 2221-2242.
Indistinguishability of the components of random spanning forests
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.
Ann. Probab., Volume 46, Number 4 (2018), 2221-2242.
Received: January 2017
Revised: August 2017
First available in Project Euclid: 13 June 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 82B43: Percolation [See also 60K35]
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