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Group-invariant percolation on graphs. (English) Zbl 0924.43002

For a closed group \(G\) of automorphisms of a graph \(X\), geometric properties such as amenability or unimodularity are related to \(G\)-invariant percolation processes. Origins of the problem may be situated in Kesten’s theorem [H. Kesten, Trans. Am. Math. Soc. 92, 336-354 (1959; Zbl 0092.33503)], insuring that a countable graph \(G\) is amenable if and only if some (every) symmetric group-invariant random walk with support generating \(G\) has spectral radius 1. Følner-type properties are used here as characterizations of amenability.
Let \(X\) be the Cayley graph of a finitely generated group \(G\). Then \(G\) is amenable if and only if, whenever \(\alpha<1\), there exists a \(G\)-invariant probability measure \(P\) on \(X\) with \(P(0\in\omega) >\alpha\) without any infinite connected components for the subgraph of \(X\) obtained by adding to \(\omega\) all edges with endpoints in \(\omega\). In order to control averaging in the nonamenable case, use is made of a mass-transport technique going back to O. Häggström [Ann. Probab. 25, 1423-1436 (1997; Zbl 0895.60098)] and W. Woess [Rend. Semin. Mat. Fis. Milano 64, 185-213 (1996; Zbl 0852.60086)]. Let \(X\) be a unimodular subgroup of \(\operatorname{Aut}(X)\) acting transitively on \(X\). Then \(X\) is amenable if and only if there exists a \(G\)-invariant nonempty connected subgraph of \(X\) with probability \(p_c\), the latter being the infimum over all probabilities \(P\) such that \(P(x\in\omega)=p\) and all \(x\in\omega\) are independent.

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
60D05 Geometric probability and stochastic geometry
60B99 Probability theory on algebraic and topological structures
60K35 Interacting random processes; statistical mechanics type models; percolation theory
31C20 Discrete potential theory
05C05 Trees
82B43 Percolation
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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