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April 2001 Loss Network Representation of Peierls Contours
Roberto Fernández, Pablo A. Ferrari, Nancy L. Garcia
Ann. Probab. 29(2): 902-937 (April 2001). DOI: 10.1214/aop/1008956697


We present a probabilistic approach for the study of systems with exclusions in the regime traditionally studied via cluster-expansion methods. In this paper we focus on its application for the gases of Peierls contours found in the study of the Ising model at low temperatures, but most of the results are general. We realize the equilibrium measure as the invariant measure of a loss network process whose existence is ensured by a subcriticality condition of a dominant branching process. In this regime the approach yields, besides existence and uniqueness of the measure, properties such as exponential space convergence and mixing, and a central limit theorem. The loss network converges exponentially fast to the equilibrium measure, without metastable traps. This convergence is faster at low temperatures, where it leads to the proof of an asymptotic Poisson distribution of contours. Our results on the mixing properties of the measure are comparable to those obtained with “duplicated-variables expansion,” used to treat systems with disorder and coupled map lattices. It works in a larger region of validity than usual cluster-expansion formalisms, and it is not tied to the analyticity of the pressure. In fact, it does not lead to any kind of expansion for the latter, and the properties of the equilibrium measure are obtained without resorting to combinatorial or complex analysis techniques.


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Roberto Fernández. Pablo A. Ferrari. Nancy L. Garcia. "Loss Network Representation of Peierls Contours." Ann. Probab. 29 (2) 902 - 937, April 2001.


Published: April 2001
First available in Project Euclid: 21 December 2001

zbMATH: 1015.60090
MathSciNet: MR1849182
Digital Object Identifier: 10.1214/aop/1008956697

Primary: 60K35 , 82B , 82C

Keywords: animal models , central limit theorem , Ising model , Loss networks , Oriented percolation , Peierls contours , Poisson approximation

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 2 • April 2001
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