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Critical behavior of percolation and Markov fields on branching planes

Part of: Equilibrium statistical mechanics Special processes

Published online by Cambridge University Press:  14 July 2016

C. Chris Wu
C. Chris Wu*
Affiliation:
Courant Institute of Mathematical Sciences, New York
*
∗ Present address: Department of Mathematics, Penn State University, Beaver Campus, Monaca, PA 15061, USA.
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Abstract

For an independent percolation model on , where is a homogeneous tree and is a one-dimensional lattice, it is shown, by verifying that the triangle condition is satisfied, that the percolation probability θ (p) is a continuous function of p at the critical point pc, and the critical exponents , γ, δ, and Δ exist and take their mean-field values. Some analogous results for Markov fields on are also obtained.

Keywords

GIBBS MEASURECRITICAL EXPONENTSPOWER LAWSMEAN-FIELD VALUESISING MODELSMAGNETIZATION

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B27: Critical phenomena
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 3 , September 1993 , pp. 538 - 547
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

Research supported in part by NSF Grants DMS-8902156 and DMS-9196086.

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