## Electronic Journal of Probability

### Stein's Method for Dependent Random Variables Occuring in Statistical Mechanics

#### Abstract

We develop Stein's method for exchangeable pairs for a rich class of distributional approximations including the Gaussian distributions as well as the non-Gaussian limit distributions. As a consequence we obtain convergence rates in limit theorems of partial sums for certain sequences of dependent, identically distributed random variables which arise naturally in statistical mechanics, in particular in the context of the Curie-Weiss models. Our results include a Berry-Esseen rate in the Central Limit Theorem for the total magnetization in the classical Curie-Weiss model, for high temperatures as well as at the critical temperature, where the Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds as the temperature converges to one and obtain a threshold for the speed of this convergence. Single spin distributions satisfying the Griffiths-Hurst-Sherman (GHS) inequality like models of liquid helium or continuous Curie-Weiss models are considered.

#### Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 30, 962-988.

Dates
Accepted: 28 June 2010
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819815

Digital Object Identifier
doi:10.1214/EJP.v15-777

Mathematical Reviews number (MathSciNet)
MR2659754

Zentralblatt MATH identifier
1225.60042

Rights

#### Citation

Eichelsbacher, Peter; Loewe, Matthias. Stein's Method for Dependent Random Variables Occuring in Statistical Mechanics. Electron. J. Probab. 15 (2010), paper no. 30, 962--988. doi:10.1214/EJP.v15-777. https://projecteuclid.org/euclid.ejp/1464819815

#### References

• Barbour, A. D. Equilibrium distributions for Markov population processes. Adv. in Appl. Probab. 12 (1980), no. 3, 591–614.
• M. Blume, V. J. Emery, and R. B. Griffiths, Ising model for the $\lambda$ transition and phase separation in ${H}e^3$–${H}e^4$ mixtures, Phys. Rev. A 4 (1971), 1071–1077.
• S. Chatterjee, J. Fulman, and A. Rollin, Exponential approximation by Stein's method and spectral graph theory, preprint, 2009.
• S. Chatterjee and Q.-M. Shao, Stein's method of exchangeable pairs with application to the Curie-Weiss model, preprint, 2009.
• Chen, Louis H. Y.; Shao, Qi-Man. Stein's method for normal approximation. An introduction to Stein's method, 1–59, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 4, Singapore Univ. Press, Singapore, 2005.
• Eichelsbacher, Peter; Löwe, Matthias. Moderate deviations for the overlap parameter in the Hopfield model. Probab. Theory Related Fields 130 (2004), no. 4, 441–472.
• Eichelsbacher, Peter; Reinert, Gesine. Stein's method for discrete Gibbs measures. Ann. Appl. Probab. 18 (2008), no. 4, 1588–1618.
• Ellis, Richard S. Concavity of magnetization for a class of even ferromagnets. Bull. Amer. Math. Soc. 81 (1975), no. 5, 925–929.
• Ellis, Richard S. Entropy, large deviations, and statistical mechanics.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 271. Springer-Verlag, New York, 1985. xiv+364 pp. ISBN: 0-387-96052-X
• Ellis, Richard S.; Monroe, James L.; Newman, Charles M. The GHS and other correlation inequalities for a class of even ferromagnets. Comm. Math. Phys. 46 (1976), no. 2, 167–182.
• Ellis, Richard S.; Newman, Charles M. Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44 (1978), no. 2, 117–139.
• Ellis, Richard S.; Newman, Charles M. Necessary and sufficient conditions for the ${\rm GHS}$ inequality with applications to analysis and probability. Trans. Amer. Math. Soc. 237 (1978), 83–99.
• Ellis, Richard S.; Newman, Charles M.; Rosen, Jay S. Limit theorems for sums of dependent random variables occurring in statistical mechanics. II. Conditioning, multiple phases, and metastability. Z. Wahrsch. Verw. Gebiete 51 (1980), no. 2, 153–169.
• R.B. Griffiths, C.A. Hurst, and S. Sherman, Concavity of magnetization of an Ising ferromagnet in a positive external field}, J. Mathematical Phys. 11 (1970), 790–795.
• Lebowitz, Joel L. GHS and other inequalities. Comm. Math. Phys. 35 (1974), 87–92.
• Rinott, Yosef; Rotar, Vladimir. On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted $U$-statistics. Ann. Appl. Probab. 7 (1997), no. 4, 1080–1105.
• Shao, Qi-Man; Su, Zhong-Gen. The Berry-Esseen bound for character ratios. Proc. Amer. Math. Soc. 134 (2006), no. 7, 2153–2159.
• Simon, Barry; Griffiths, Robert B. The $(\phi \sp{4})\sb{2}$ field theory as a classical Ising model. Comm. Math. Phys. 33 (1973), 145–164.
• Simon, Barry; Griffiths, Robert B. The $(\phi \sp{4})\sb{2}$ field theory as a classical Ising model. Comm. Math. Phys. 33 (1973), 145–164.
• Stein, Charles. Approximate computation of expectations.Institute of Mathematical Statistics Lecture Notes–-Monograph Series, 7. Institute of Mathematical Statistics, Hayward, CA, 1986. iv+164 pp. ISBN: 0-940600-08-0
• Stein, Charles; Diaconis, Persi; Holmes, Susan; Reinert, Gesine. Use of exchangeable pairs in the analysis of simulations. Stein's method: expository lectures and applications, 1–26, IMS Lecture Notes Monogr. Ser., 46, Inst. Math. Statist., Beachwood, OH, 2004.
• Thompson, Colin J. Mathematical statistical mechanics.A Series of Books in Applied Mathematics.The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1972. x+278 pp.