Indistinguishability of Percolation Clusters

Details Indistinguishability of Percolation Clusters Indistinguishability of Percolation Clusters

1998-11-29

arxiv.org/abs/math/9811170v2

AnnalsProbab.27:1809-1836,1999

Mathematics

Probability

To appear in Ann. Probab

Scientific paper

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 $p_u$.

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