Lower bounds on the cluster size distribution |
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| Authors: | Michael Aizenman FranÇois Delyon Bernard Souillard |
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| Affiliation: | (1) IHES, Bures-sur-Yvette, France;(2) Present address: Physics Department, Princeton University, Princeton, New Jersey;(3) Centre de Physique Theorique (Equipe de Recherche du CNRS No. 174), Ecole Polytechnique, Palaiseau, France |
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| Abstract: | We rigorously prove that the probabilityPn that the origin of ad-dimensional lattice belongs to a cluster of exactlyn sites satisfiesPn > exp(– n(d–1)/d) whenever percolation occurs. This holds for the usual (noninteracting) percolation models for any concentrationp > pc, as well as for the equilibrium states of lattice spin systems with quite general interactions. Such a lower bound applies also if no percolation occurs, but if it appears in some other phase of the system. |
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| Keywords: | Percolation Gibbs states cluster size distribution nucleation stochastic geometry |
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