Indistinguishability of percolation clusters (Q1577738)

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.