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.

discrete moment problems

statistical mechanics

infinite exchangeability

Author

Thomas Liggett

University of California

Jeffrey Steif

Chalmers, Mathematical Sciences

University of Gothenburg

Balint Toth

Budapest University of Technology and Economics

Annals of Probability

0091-1798 (ISSN)

Vol. 35 3 867-914

Subject Categories

Other Mathematics

DOI

10.1214/009117906000001033

More information

Latest update

3/19/2018