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Indistinguishability of Percolation Clusters

Russell Lyons·Oded Schramm
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Abstract

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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Russell Lyons, Oded Schramm. 1999-05-02. Indistinguishability of Percolation Clusters. https://arxiv.org/abs/math/9811170

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