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May 2007 Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem
Thomas M. Liggett, Jeffrey E. Steif, Bálint Tóth
Ann. Probab. 35(3): 867-914 (May 2007). DOI: 10.1214/009117906000001033

Abstract

We show that a large collection of statistical mechanical systems with quadratically represented Hamiltonians on the complete graph can be extended to infinite exchangeable processes. This extends a known result for the ferromagnetic Curie–Weiss Ising model and includes as well all ferromagnetic Curie–Weiss Potts and Curie–Weiss Heisenberg models. By de Finetti’s theorem, this is equivalent to showing that these probability measures can be expressed as averages of product measures. We provide examples showing that “ferromagnetism” is not however in itself sufficient and also study in some detail the Curie–Weiss Ising model with an additional 3-body interaction. Finally, we study the question of how much the antiferromagnetic Curie–Weiss Ising model can be extended. In this direction, we obtain sharp asymptotic results via a solution to a new moment problem. We also obtain a “formula” for the extension which is valid in many cases.

Citation

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Thomas M. Liggett. Jeffrey E. Steif. Bálint Tóth. "Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem." Ann. Probab. 35 (3) 867 - 914, May 2007. https://doi.org/10.1214/009117906000001033

Information

Published: May 2007
First available in Project Euclid: 10 May 2007

MathSciNet: MR2319710
zbMATH: 1126.44007
Digital Object Identifier: 10.1214/009117906000001033

Subjects:
Primary: 44A60, 60G09, 60K35, 82B20

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 3 • May 2007
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