  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 $p_u$.
