The Annals of Probability

Replica symmetry breaking and exponential inequalities for the Sherrington-Kirkpatrick model

Michel Talagrand

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Abstract

We provide an extremely accurate picture of the Sherrington – Kirkpatrick model in three cases:for high temperature, for large external field and for any temperature greater than or equal to 1 and sufficiently small external field. We describe the system at the level of the central limit theorem, or as physicists would say, at the level of fuctuations around the mean field. We also obtain much more detailed information, in the form of exponential inequalities that express a uniform control over higher order moments.We give a complete, rigorous proof that at the generic point of the predicted low temperature region there is “replica symmetry breaking,” in the sense that the system is unstable with respect to an infinitesimal coupling between two replicas.

Article information

Source
Ann. Probab., Volume 28, Number 3 (2000), 1018-1062.

Dates
First available in Project Euclid: 18 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1019160325

Digital Object Identifier
doi:10.1214/aop/1019160325

Mathematical Reviews number (MathSciNet)
MR1797303

Zentralblatt MATH identifier
1034.82027

Subjects
Primary: 82D30: Random media, disordered materials (including liquid crystals and spin glasses)
Secondary: 60G15: Gaussian processes 60G70: Extreme value theory; extremal processes

Keywords
Disorder mean field

Citation

Talagrand, Michel. Replica symmetry breaking and exponential inequalities for the Sherrington-Kirkpatrick model. Ann. Probab. 28 (2000), no. 3, 1018--1062. doi:10.1214/aop/1019160325. https://projecteuclid.org/euclid.aop/1019160325


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References

  • [1] Aizenman, M., Lebowitz, J. and Ruelle, D. (1981). Some rigorous results on the Sherrington-Kirkpatrick model. Comm. Math. Phys. 112 3-20.
  • [2] Comets, F. and Neveu, J. (1995). The Sherrington-Kirkpatrick model of spin-glasses and stochastic calculus: the high temperature case. Comm. Math. Phys. 166 549-564.
  • [3] M´ezard, M., Parisi, G. and Virasiro, M. (1987). Spin Glass Theory and Beyond. World Scientific, Singapore.
  • [4] Shcherbina, M. (1997). On the replica-symmetric solution for the Sherrington-Kirkpatrick model. Helv. Phys. Acta. 70 838-853.
  • [5] Talagrand, M. (1998). The Sherrington-Kirkpatrick model: a challenge to mathematicians. Probab. Theory Related Fields 110 109-176.
  • [6] Talagrand, M. (1998). Rigorous results for the Hopfield Model with many patterns. Probab. Theory Related Fields 110 177-286.
  • [7] Talagrand, M. (1998). Huge random structures and mean field models for spin glasses. In Proceedings of the International Congress of Mathematicians, Documenta Math. Extra volume I.
  • [8] Talagrand, M. (1999). Verres de spin et optimisation combinatoire. S´eminaire Bourbaki, Ast´erisque. To appear.
  • [9] Talagrand, M. (2000). Probability theory and spin glasses. Unpublished manuscript.