The Annals of Probability

Indistinguishability of Percolation Clusters

Russell Lyons and Oded Schramm

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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 nondecay of connectivity (a.k.a. long-range order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products and inequalities for $p_u$.

Article information

Ann. Probab. Volume 27, Number 4 (1999), 1809-1836.

First available in Project Euclid: 31 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B43: Percolation [See also 60K35] 60B99: None of the above, but in this section
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Finite energy Cayley graph group Kazhdan wreath product uniqueness connectivity transitive nonamenable.


Lyons, Russell; Schramm, Oded. Indistinguishability of Percolation Clusters. Ann. Probab. 27 (1999), no. 4, 1809--1836. doi:10.1214/aop/1022874816.

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