2005/12/31 by Thomas M. Liggett, Jeffrey E. Steif, Bálint Tóth · 1 citation
Mathematics · Physics and Astronomy · #Ferromagnetism #Infinite product #Ising model #Markov Chains and Monte Carlo Methods #Moment (physics) #Potts model #Product (mathematics) #Quadratic growth #Statistical Mechanics and Entropy #Statistical mechanics #Statistical model #Theoretical and Computational Physics #math.PR #msc:44A60 #msc:60G09 #msc:60K35 #msc:82B20
paper · pdf · doi:10.1214/009117906000001033
published as Annals of Probability 2007, Vol. 35, No. 3, 867-914 · Published at http://dx.doi.org/10.1214/009117906000001033 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2007/05/01 · arxiv created 2007/07/30 · arxiv updated 2011/11/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.