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
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
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