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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.)
##### Keywords:
percolation; abelian groups; graph limits; locality
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