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

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.