Tree graph inequalities and critical behavior in percolation models

Michael Aizenman, Charles M. Newman

Research output: Contribution to journalArticle

133 Scopus citations

Abstract

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
Volume36
Issue number1-2
DOIs
StatePublished - Jul 1 1984
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Keywords

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

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