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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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.

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

First available in Project Euclid: 10 May 2007

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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

Statistical mechanics infinite exchangeability discrete moment problems


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.

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