## The Annals of Probability

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

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 35, Number 3 (2007), 867-914.

Dates
First available in Project Euclid: 10 May 2007

https://projecteuclid.org/euclid.aop/1178804317

Digital Object Identifier
doi:10.1214/009117906000001033

Mathematical Reviews number (MathSciNet)
MR2319710

Zentralblatt MATH identifier
1126.44007

#### Citation

Liggett, Thomas M.; Steif, Jeffrey E.; Tóth, Bálint. Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem. Ann. Probab. 35 (2007), no. 3, 867--914. doi:10.1214/009117906000001033. https://projecteuclid.org/euclid.aop/1178804317

#### References

• Akritas, A. G., Akritas, E. K. and Malaschonok, G. I. (1996). Various proofs of Sylvester's (determinant) identity. Math. Comput. Simulation 42 585--593.
• Aldous, D. J. (1985). Exchangeability and related topics. École d'été de probabilités de Saint-Flour, XIII. Lecture Notes in Math. 1117. 1--198. Springer, Berlin.
• Cramér, H. (1962). Random Variables and Probability Distributions, 2nd ed. Cambridge Univ. Press.
• Diaconis, P. (1988). Recent progress on de Finetti's notions of exchangeability. In Bayesian Statistics 3 111--125. Oxford Univ. Press.
• Diaconis, P. and Freedman, D. (1980). Finite exchangeable sequences Ann. Probab. 8 745--764.
• Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Springer, New York.
• Fristedt, B. and Gray, L. (1997). A Modern Approach to Probability Theory. Birkhäuser, Boston.
• Fröhlich, J. and Spencer, T. (1981). The Kosterlitz--Thouless transition in two-dimensional Abelian spin systems and the Coulomb gas. Comm. Math. Phys. 81 527--602.
• Gantmacher, F. R. (1959). The Theory of Matrices 1. Chelsea, New York.
• Häggström, O. (1999). Positive correlations in the fuzzy Potts model. Ann. Appl. Probab. 9 1149--1159.
• Kac, M. (1968). Mathematical mechanisms of phase transition. In Statistical Physics: Phase Transitions and Superfluidity (M. Chretien, E. P. Gross, and S. Deser eds.) 1 241--305. Gordon and Breach, New York.
• Karlin, S. (1968). Total Positivity 1. Stanford Univ. Press.
• Karlin, S. and Shapley, L. (1953). Geometry of moment spaces. Mem. Amer. Math. Soc. 1953 93.
• Karlin, S. and Studden, W. J. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics. Wiley, New York.
• Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.
• Liggett, T. M. and Steif, J. E. (2006). Stochastic domination: The contact process, Ising models and FKG measures. Ann. Inst. H. Poincaré Probab. Statist. 42 223--243.
• Lindsay, B. (1989). On the determinants of moment matrices. Ann. Statist. 17 711--721.
• Mehta, M. L. (2004). Random Matrices, 3rd ed. Academic Press, Amsterdam.
• Olivieri, E. and Vares, M. E. (2005). Large Deviations and Metastability. Cambridge Univ. Press.
• Papangelou, F. (1989). On the Gaussian fluctuations of the critical Curie--Weiss model in statistical mechanics. Probab. Theory Related Fields 83 265--278.
• Scarsini, M. (1985). Lower bounds for the distribution function of a $k$-dimensional $n$-extendible exchangeable process. Statist. Probab. Lett. 3 57--62.
• Shohat, J. A. and Tamarkin, J. D. (1943). The Problem of Moments. Amer. Math. Soc., New York.
• Spizzichino, F. (1982). Extendibility of symmetric probability distributions and related bounds. In Exchangeability in Probability and Statistics (Rome, 1981) 313--320. North-Holland, Amsterdam.
• Tagliani, A. (1999). Hausdorff moment problem and maximum entropy: A unified approach Appl. Math. Comput. 105 291--305.