Journal of Statistical Physics

, Volume 36, Issue 1, pp 107–143

Tree graph inequalities and critical behavior in percolation models

  • Michael Aizenman
  • Charles M. Newman
Articles

DOI: 10.1007/BF01015729

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

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 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

Authors and Affiliations

  • Michael Aizenman
    • 1
  • Charles M. Newman
    • 2
  1. 1.Departments of Mathematics and PhysicsRutgers UniversityNew BrunswickNew Jersey
  2. 2.Department of MathematicsUniversity of ArizonaTucson
  3. 3.Department of Theoretical MathematicsThe Weizmann Institute of ScienceRehovotIsrael
  4. 4.Institute of Mathematics and Computer ScienceThe Hebrew UniversityJerusalemIsrael