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Lower bounds on the cluster size distribution

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Abstract

We rigorously prove that the probabilityP n that the origin of ad-dimensional lattice belongs to a cluster of exactlyn sites satisfiesP n > exp(−αn (d−1)/d) whenever percolation occurs. This holds for the usual (noninteracting) percolation models for any concentrationp > p c , 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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Author notes

  1. Michael Aizenman

    Present address: Physics Department, Princeton University, Princeton, New Jersey

Authors and Affiliations

  1. IHES, Bures-sur-Yvette, France

    Michael Aizenman

  2. Centre de Physique Theorique (Equipe de Recherche du CNRS No. 174), Ecole Polytechnique, Palaiseau, France

    FranÇois Delyon & Bernard Souillard

Authors
  1. Michael Aizenman
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  2. FranÇois Delyon
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  3. Bernard Souillard
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Aizenman, M., Delyon, F. & Souillard, B. Lower bounds on the cluster size distribution. J Stat Phys 23, 267–280 (1980). https://doi.org/10.1007/BF01011369

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  • DOI: https://doi.org/10.1007/BF01011369

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