The Annals of Probability

Loss Network Representation of Peierls Contours

Roberto Fernández, Pablo A. Ferrari, and Nancy L. Garcia

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Abstract

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.

Article information

Source
Ann. Probab., Volume 29, Number 2 (2001), 902-937.

Dates
First available in Project Euclid: 21 December 2001

Permanent link to this document
https://projecteuclid.org/euclid.aop/1008956697

Digital Object Identifier
doi:10.1214/aop/1008956697

Mathematical Reviews number (MathSciNet)
MR1849182

Zentralblatt MATH identifier
1015.60090

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B 82C

Keywords
Peierls contours animal models loss networks Ising model oriented percolation central limit theorem Poisson approximation

Citation

Fernández, Roberto; Ferrari, Pablo A.; Garcia, Nancy L. Loss Network Representation of Peierls Contours. Ann. Probab. 29 (2001), no. 2, 902--937. doi:10.1214/aop/1008956697. https://projecteuclid.org/euclid.aop/1008956697


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  • IMECC, UNICAMP Caixa Postal 6065 13081-970 Campinas SP Brazil E-mail: nancy@ime.unicamp.br