Celebratio Mathematica — Liggett — Complete Bibliography

Complete Bibliography

Works connected to Bálint Tóth

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T. M. Liggett, J. E. Steif, and B. Tóth: “Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem,” Ann. Probab. 35 : 3 (2007), pp. 867–914. MR 2319710 Zbl 1126.44007 ArXiv math/0512191 article

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.

Jeffrey Edward Steif	Related
Bálint Tóth	Related