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Locality of percolation for abelian Cayley graphs. (English) Zbl 1388.60165
In this article, the authors consider bond percolation on Cayley graphs of abelian groups. For any graph \(G\), let \(p_c(G)\) be the critical probability. Consider a sequence of Cayley graphs \(\mathcal{G}_n\) which converge to a Cayley graph \(\mathcal{G}\). The authors prove that \(p_c(\mathcal{G}_n) \rightarrow p_c(\mathcal{G})\) under the assumption that \(\sup_n p_c(\mathcal{G}_n) < 1\). This proves a special case of a conjecture of O. Schramm. The general conjecture claimed an analogous result for vertex-transitive graphs.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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