Abstract
We rigorously investigate the size dependence of disordered mean-field models with finite local spin space in terms of metastates. Thereby we provide an illustration of the framework of metastates for systems of randomly competing Gibbs measures. In particular we consider the thermodynamic limit of the empirical metastate\(1/N\sum\nolimits_{n - 1}^N {\delta _{\mu _\eta (\eta )} } \), whereμ n (η) is the Gibbs measure in the finite volume {1,…,n} and the frozen disorder variableη is fixed. We treat explicitly the Hopfield model with finitely many patterns and the Curie-Weiss random field Ising model. In both examples in the phase transition regime the empirical metastate is dispersed for largeN. Moreover, it does not converge for a.e.η, but rather in distribution, for whose limits we given explicit expressions. We also discuss another notion of metastates, due to Aizenman and Wehr.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. M. G. Amaro de Matos, A. E. Patrick, and V. A. Zagrebnov, Random Infinite-Volume Gibbs States for the Curie-Weiss Random Field Ising Model,J. Stat. Phys. 66:139–164 (1992).
J. M. G. Amaro de Matos and J. F. Perez, Fluctuations in the Curie-Weiss Version of the Random Field Ising Model,J. Stat. Phys. 62:587–608 (1990).
M. Aizenman and J. Wehr, Rounding Effects of Quenched Randomness on First-Order Phase Transitions,Comm. Math. Phys. 130:489–528 (1990).
A. Bovier and V. Gayrard, The retrieval phase of the Hopfield model: A rigorous analysis of the overlap distribution, to appear inProb. Theor. Rel. Fields (1995).
A. Bovier and V. Gayrard, Hopfield models as a generalized random mean field model, WIAS preprint 253, Berlin (1996), to appear in “Mathematics of spin glasses and neural networks,” A. Bovier and P. Picco, eds., “Progress in Probability,” Birkhäuser (1997).
A. Bovier, V. Gayrard, and P. Picco, Gibbs states of the Hopfield model in the regime of perfect memory,Prob. Theor. Rel. Fields 100:329–363 (1994); Gibbs states of the Hopfield model with extensively many patterns,J. Stat. Phys. 79:395–414 (1995).
F. Comets, Large Deviation Estimates for a Conditional Probability Distribution. Applications to Random Interaction Gibbs Measures,Prob. Th. Rel. Fields 80:407–432 (1989).
J.-D. Deuschel and D. W. Stroock,Large Deviations, Pure and Applied Mathematics, vol. 137, Academic Press, Boston (1989).
A. Dembo and O. Zeitouni,Large Deviations Techniques, Jones and Bartlett, Boston, London (1993).
R. S. Ellis, Entropy,Large Deviations, and Statistical Mechanics, Grundlehren der mathematischen Wissenschaften, vol. 271, Springer, New York.
W. Feller,An Introduction yo Probability Theory and its Applications, Wiley, New York, London, Sidney, (1966).
B. Gentz, An almost sure Central Limit Theorem for the overlap parameters in the Hopfield model,Stochastic Process. Appl. 62, no. 2, 243–262 (1996).
H. O. Georgii,Gibbs measures and phase transitions. Studies in mathematics, vol. 9 (de Gruyter, Berlin, New York, 1988).
J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities,Proc. Natl. Acad. Sci. USA 79:2554–2558 (1982).
F. Ledrappier, Pressure and Variational Principle for Random Ising Model,Comm. Math. Phys. 56:297–302 (1977).
J. T. Lewis, C. E. Pfister, and W. G. Sullivan, Entropy, Concentration of Probability and Conditional Limit Theorems,Markov Proc. Rel. Felds 1:319–386 (1995).
C. M. Newman,Topics in Disordered Systems, to appear in “ETH Lecture Notes Series,” Birkhäuser (1996).
C. M. Newman and D. L. Stein, Chaotic Size Dependence in Spin Glasses, inCellular Automata and Cooperative Systems, Boccara, Goles, Martinez, Picco (Eds.), Nato ASI Series C Vol. 396, Kluwer, Dordrecht (1993).
C. M. Newman and D. L. Stein, Non-Mean Field Behavior in realistic Spin glasses,Phys. Rev. Lett. 76, No. 3, 515 (1996).
C. M. Newman and D. L. Stein, Spatial Inhomogeneity and thermodynamic chaos,Phys. Rev. Lett. 76, No. 25, 4821 (1996).
G. Parisi, Recent rigorous results support the predictions of spontaneously broken replica symmetriy for realistic spin glass, preprint, March, 1996. Available as condmat preprint 9603101 at http://www.sissa.it.
E. Rio, Strong Approximation for set-indexed partial-sum processes, via KMT constructions II,Ann. Prob. 21, No. 3, 1706–1727 (1993).
T. Seppäläinen, Entropy, limit theorems, and variational principles for disordered lattice systems,Commun. Math. Phys. 171:233–277 (1995).
S. R. Salinas and W. F. Wreszinski, On the Mean-Field Ising Model in a Random External Field,J. Stat. Phys. 41:299–313 (1985).
M. Ledoux, M. Talagrand,Probability in Banach Spaces, Springer Verlag (1991).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Külske, C. Metastates in disordered mean-field models: Random field and hopfield models. J Stat Phys 88, 1257–1293 (1997). https://doi.org/10.1007/BF02732434
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02732434