We show that when percolation produces infinitely many infinite 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 (a.k.a. long-range order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products and inequalities for $p_u$.
"Indistinguishability of Percolation Clusters." Ann. Probab. 27 (4) 1809 - 1836, October 1999. https://doi.org/10.1214/aop/1022874816