Indistinguishability of Percolation Clusters

Russell Lyons, Oded Schramm

Published 1998-11-29, updated 1999-05-02Version 2

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 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 $p_u$.