2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/175540We 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 non-decay of connectivity (a.k.a. long-range order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products, and inequalities for $p_u$.To appear in Ann. ProbabProbabilityMathematical Physics82B43, 60B99 (Primary), 60K35, 60D05 (Secondary)text