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Indistinguishability of percolation clusters. (English) Zbl 0960.60013
The authors show that when percolation produces infinitely many clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity (long-range order of the process). The authors then derive applications concerning uniqueness in Kazhdan groups and in wreath products and inequalities for \(p_u\) (the infimum of \(p\) in the uniqueness phase). Crucial techniques for the proofs are the mass-transport principle and stationarity of delayed simple random walk.

MSC:
60B99 Probability theory on algebraic and topological structures
82B43 Percolation
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry
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