Journal of Statistical Physics

, Volume 23, Issue 3, pp 267–280 | Cite as

Lower bounds on the cluster size distribution

  • Michael Aizenman
  • FranÇois Delyon
  • Bernard Souillard


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.

Key words

Percolation Gibbs states cluster size distribution nucleation stochastic geometry 


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Copyright information

© Plenum Publishing Corporation 1980

Authors and Affiliations

  • Michael Aizenman
    • 1
  • FranÇois Delyon
    • 2
  • Bernard Souillard
    • 2
  1. 1.IHESBures-sur-YvetteFrance
  2. 2.Centre de Physique Theorique (Equipe de Recherche du CNRS No. 174)Ecole PolytechniquePalaiseauFrance

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