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Percolation on Grigorchuk groups. (English) Zbl 1042.20035
Summary: Let \(p_c(G)\) be the critical probability of the site percolation on the Cayley graph of group \(G\). I. Benjamini and O. Schramm [Electron. Commun. Probab. 1, 71-82 (1996; Zbl 0890.60091)] conjectured that \(p_c<1\), given the group is infinite and not a finite extension of \(\mathbb{Z}\). The conjecture was proved earlier for groups of polynomial and exponential growth and remains open for groups of intermediate growth. In this note. we prove the conjecture for a special class of Grigorchuk groups, which is a special class of groups of intermediate growth. The proof is based on an algebraic construction. No previous knowledge of percolation is assumed.

MSC:
20F69 Asymptotic properties of groups
60G50 Sums of independent random variables; random walks
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C30 Enumeration in graph theory
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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