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Percolation on nonunimodular transitive graphs. (English) Zbl 1114.60083

Summary: We extend some of the fundamental results about percolation on unimodular nonamenable graphs to nonunimodular graphs. We show that they cannot have infinitely many infinite clusters at critical Bernoulli percolation. In the case of heavy clusters, this result has already been established, but it also follows from one of our results. We give a general necessary condition for nonunimodular graphs to have a phase with infinitely many heavy clusters. We present an invariant spanning tree with \(p_c=1\) on some nonunimodular graph. Such trees cannot exist for nonamenable unimodular graphs. We show a new way of constructing nonunimodular graphs that have properties more peculiar than the ones previously known.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
60B99 Probability theory on algebraic and topological structures
60C05 Combinatorial probability
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