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

• 成果类型: Article
• 署名作者: Liggett, Thomas M.; Steif, Jeffrey E.; Toth, Balint
• 署名单位: University of California System; University of California Los Angeles; Chalmers University of Technology; University of Gothenburg
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000001033
• 发表日期: 2007
• 页码: 867-914
• 关键词: model

摘要

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.