The Annals of Probability

Locality of percolation for Abelian Cayley graphs

Abstract

We prove that the value of the critical probability for percolation on an Abelian Cayley graph is determined by its local structure. This is a partial positive answer to a conjecture of Schramm: the function $\mathrm{p}_{\mathrm{c}}$ defined on the set of Cayley graphs of Abelian groups of rank at least $2$ is continuous for the Benjamini–Schramm topology. The proof involves group-theoretic tools and a new block argument.

Article information

Source
Ann. Probab., Volume 45, Number 2 (2017), 1247-1277.

Dates
Revised: March 2015
First available in Project Euclid: 31 March 2017

https://projecteuclid.org/euclid.aop/1490947319

Digital Object Identifier
doi:10.1214/15-AOP1086

Mathematical Reviews number (MathSciNet)
MR3630298

Zentralblatt MATH identifier
06797091

Citation

Martineau, Sébastien; Tassion, Vincent. Locality of percolation for Abelian Cayley graphs. Ann. Probab. 45 (2017), no. 2, 1247--1277. doi:10.1214/15-AOP1086. https://projecteuclid.org/euclid.aop/1490947319

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