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Uniqueness and multiplicity of infinite clusters. (English) Zbl 1125.82005
Denteneer, Dee (ed.) et al., Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference ‘Dynamical systems, probability theory, and statistical mechanics’, Eindhoven, The Netherlands, January 3–7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-64-1/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 24-36 (2006).
Summary: 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.
For the entire collection see [Zbl 1113.60008].

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry
82B43 Percolation
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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