Benjamini, I.; Lyons, R.; Peres, Y.; Schramm, O. Group-invariant percolation on graphs. (English) Zbl 0924.43002 Geom. Funct. Anal. 9, No. 1, 29-66 (1999). 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. Reviewer: Jean-Paul Pier (Luxembourg) Cited in 2 ReviewsCited in 83 Documents 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 Keywords:trees; graph; amenability; percolation; random walk Citations:Zbl 0092.33503; Zbl 0895.60098; Zbl 0852.60086 PDFBibTeX XMLCite \textit{I. Benjamini} et al., Geom. Funct. Anal. 9, No. 1, 29--66 (1999; Zbl 0924.43002) Full Text: DOI