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 language | English (US) |
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Pages (from-to) | 107-143 |
Number of pages | 37 |
Journal | Journal of Statistical Physics |
Volume | 36 |
Issue number | 1-2 |
DOIs | |
State | Published - Jul 1984 |
Externally published | Yes |
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