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March 2017 Locality of percolation for Abelian Cayley graphs
Sébastien Martineau, Vincent Tassion
Ann. Probab. 45(2): 1247-1277 (March 2017). DOI: 10.1214/15-AOP1086

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.

Citation

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Sébastien Martineau. Vincent Tassion. "Locality of percolation for Abelian Cayley graphs." Ann. Probab. 45 (2) 1247 - 1277, March 2017. https://doi.org/10.1214/15-AOP1086

Information

Received: 1 January 2014; Revised: 1 March 2015; Published: March 2017
First available in Project Euclid: 31 March 2017

zbMATH: 06797091
MathSciNet: MR3630298
Digital Object Identifier: 10.1214/15-AOP1086

Subjects:
Primary: 60K35, 82B43
Secondary: 05C25, 82B20

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 2 • March 2017
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