Tree graph inequalities and critical behavior in percolation models |
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Authors: | Michael Aizenman Charles M Newman |
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Institution: | (1) Departments of Mathematics and Physics, Rutgers University, 08903 New Brunswick, New Jersey;(2) Department of Mathematics, University of Arizona, 85721 Tucson, Arizona;(3) Present address: Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot, Israel;(4) Present address: Institute of Mathematics and Computer Science, The Hebrew University, Jerusalem, Israel |
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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 associated with the expected cluster sizex and the structure of then-site connection probabilities = n(x1,..., xn). It is shown that quite generally ![gamma](/content/u1312238l8884077/xxlarge947.gif) 1. The upper critical dimension, above which 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 (x, y)=O(¦x -y¦–(d–2+ ), atp=p
c, our criterion shows that =1 if > (6-d)/3. The connectivity functions n are generally bounded by tree diagrams which involve the two-point function. We conjecture that above the critical dimension the asymptotic behavior of 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 2
(x, y).
A. P. Sloan Foundation Research Fellow. Research supported in part by the National Science Foundation Grant No. PHY-8301493.Research supported in part by the National Science Foundation Grant No. MCS80-19384. |
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Keywords: | Percolation critical exponents correlation functions connectivity inequalities upper critical dimension cluster size distribution rigorous results |
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