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.