# Tree graph inequalities and critical behavior in percolation models

- Received:

DOI: 10.1007/BF01015729

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

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## 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=τ_{n}(x_{1},..., x_{n}). 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=p*_{c}, our criterion shows that γ=1 if η> (6-*d)/3.* The connectivity functions τ_{n} are generally bounded by tree diagrams which involve the two-point function. We conjecture that above the critical dimension the asymptotic behavior of τ_{n}, 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 τ_{2}*(x, y).*