Tree graph inequalities and critical behavior in percolation models

Michael Aizenman, Charles M. Newman

Research output: Contribution to journalArticlepeer-review


Various inequalities are derived and used for the study of the critical behavior in independent percolation models. In particular, we consider the critical exponent γ associated with the expected cluster size x and the structure of the n-site connection probabilities τ=τn(x1,..., xn). It is shown that quite generally γ≥ 1. The upper critical dimension, above which γ attains the Bethe lattice value 1, is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition. For homogeneous d-dimensional lattices with τ(x, y)=O(|x -y|-(d-2+η), at p=pc, our criterion shows that γ=1 if η> (6-d)/3. The connectivity functions τn are generally bounded by tree diagrams which involve the two-point function. We conjecture that above the critical dimension the asymptotic behavior of τn, in the critical regime, is actually given by such tree diagrams modified by a nonsingular vertex factor. Other results deal with the exponential decay of the cluster-size distribution and the function τ2(x, y).

Original languageEnglish (US)
Pages (from-to)107-143
Number of pages37
JournalJournal of Statistical Physics
Issue number1-2
StatePublished - Jul 1984


  • Percolation
  • cluster size distribution
  • connectivity inequalities
  • correlation functions
  • critical exponents
  • rigorous results
  • upper critical dimension

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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