Martineau, Sébastien; Tassion, Vincent Locality of percolation for abelian Cayley graphs. (English) Zbl 1388.60165 Ann. Probab. 45, No. 2, 1247-1277 (2017). 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. Reviewer: Arvind Ayyer (Bangalore) Cited in 17 Documents 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.) Keywords:percolation; abelian groups; graph limits; locality PDFBibTeX XMLCite \textit{S. Martineau} and \textit{V. Tassion}, Ann. Probab. 45, No. 2, 1247--1277 (2017; Zbl 1388.60165) Full Text: DOI arXiv Euclid