Hopfield models as generalized random mean field models

Abstract

We give a comprehensive self-contained review on the rigorous analysis of the thermodynamics of a class of random spin systems of mean field type whose most prominent example is the Hopfield model. We focus on the low temperature phase and the analysis of the Gibbs measures with large deviation techniques. There is a very detailed and complete picture in the regime of ``small $\a$''; a particularly satisfactory result concerns a non-trivial regime of parameters in which we prove 1) the convergence of the local ``mean fields'' to gaussian random variables with constant variance and random mean; the random means are from site to site independent gaussians themselves; 2) ``propagation of chaos'', i.e. factorization of the extremal infinite volume Gibbs measures, and 3) the correctness of the ``replica symmetric solution'' of Amit, Gutfreund and Sompolinsky [AGS]. This last result was first proven by M. Talagrand [T4], using different techniques.