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Replica symmetry breaking and exponential inequalities for the Sherrington-Kirkpatrick model. (English) Zbl 1034.82027

Summary: 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 fluctuations 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.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60G15 Gaussian processes
60G17 Sample path properties
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)

Keywords:

disorder; mean field
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References:

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