The Annals of Probability

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

Michel Talagrand

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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

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

First available in Project Euclid: 18 April 2002

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Zentralblatt MATH identifier

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

Disorder mean field


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.

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