Institute of Mathematical Statistics Lecture Notes - Monograph Series

Uniqueness and multiplicity of infinite clusters

Geoffrey Grimmett

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The Burton-Keane theorem for the almost-sure uniqueness of infinite clusters is a landmark of stochastic geometry. Let $\mu$ be a translation-invariant probability measure with the finite-energy property on the edge-set of a $d$-dimensional lattice. The theorem states that the number $I$ of infinite components satisfies $\mu(I\in\{0,1\}) = 1$. The proof is an elegant and minimalist combination of zero-one arguments in the presence of amenability. The method may be extended (not without difficulty) to other problems including rigidity and entanglement percolation, as well as to the Gibbs theory of random-cluster measures, and to the central limit theorem for random walks in random reflecting labyrinths. It is a key assumption on the underlying graph that the boundary/volume ratio tends to zero for large boxes, and the picture for non-amenable graphs is quite different.

Chapter information

Dee Denteneer, Frank den Hollander, Evgeny Verbitskiy, eds., Dynamics & Stochastics: Festschrift in honor of M. S. Keane (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006), 24-36

First available in Project Euclid: 28 November 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B43: Percolation [See also 60K35]

percolation stochastic geometry rigidity entanglement random cluster model random labyrinth

Copyright © 2006, Institute of Mathematical Statistics


Grimmett, Geoffrey. Uniqueness and multiplicity of infinite clusters. Dynamics & Stochastics, 24--36, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000040.

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