The Annals of Probability

Locality of percolation for Abelian Cayley graphs

Sébastien Martineau and Vincent Tassion

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65]

Percolation Abelian groups graph limits locality


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

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