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

Journal article, 2007

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 includes all ferromagnetic Ising, Potts and 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 Ising model with an additional 3-body interaction. Finally, we study the question of how much the antiferromagnetic 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.