The Annals of Probability

Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem

Thomas M. Liggett, Jeffrey E. Steif, and Bálint Tóth

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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.

Article information

Source
Ann. Probab., Volume 35, Number 3 (2007), 867-914.

Dates
First available in Project Euclid: 10 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1178804317

Digital Object Identifier
doi:10.1214/009117906000001033

Mathematical Reviews number (MathSciNet)
MR2319710

Zentralblatt MATH identifier
1126.44007

Subjects
Primary: 44A60: Moment problems 60G09: Exchangeability 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
Statistical mechanics infinite exchangeability discrete moment problems

Citation

Liggett, Thomas M.; Steif, Jeffrey E.; Tóth, Bálint. Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem. Ann. Probab. 35 (2007), no. 3, 867--914. doi:10.1214/009117906000001033. https://projecteuclid.org/euclid.aop/1178804317


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