Lower bounds on the cluster size distribution

Michael Aizenman, Fran Çois Delyon, Bernard Souillard

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


We rigorously prove that the probability Pn that the origin of a d-dimensional lattice belongs to a cluster of exactly n sites satisfies Pn > exp(-αn(d-1)/d) whenever percolation occurs. This holds for the usual (noninteracting) percolation models for any concentration p > 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.

Original languageEnglish (US)
Pages (from-to)267-280
Number of pages14
JournalJournal of Statistical Physics
Issue number3
StatePublished - Sep 1980
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Gibbs states
  • Percolation
  • cluster size distribution
  • nucleation
  • stochastic geometry


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