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

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

2005-12-09

arxiv.org/abs/math/0512191v4

Annals of Probability 2007, Vol. 35, No. 3, 867-914

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000001033 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117906000001033

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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