Journal of Statistical Physics

, Volume 36, Issue 1, pp 107–143

Tree graph inequalities and critical behavior in percolation models


  • Michael Aizenman
    • Departments of Mathematics and PhysicsRutgers University
  • Charles M. Newman
    • Department of MathematicsUniversity of Arizona

DOI: 10.1007/BF01015729

Cite this article as:
Aizenman, M. & Newman, C.M. J Stat Phys (1984) 36: 107. doi:10.1007/BF01015729


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 sizex and the structure of then-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 homogeneousd-dimensional lattices with τ(x, y)=O(¦x -y¦−(d−2+η), atp=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).

Key words

Percolationcritical exponentscorrelation functionsconnectivity inequalitiesupper critical dimensioncluster size distributionrigorous results

Copyright information

© Plenum Publishing Corporation 1984