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 α”; 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.

Abstract: We prove a large deviation principle for the finite dimensional marginals of the Gibbs distribution of the macroscopic ‘overlap’-parameters in the Hopfield model in the case where the number of random ‘patterns’, M, as a function of the system size N satisfies lim supM(N)/N = 0. In this case the rate function is independent of the disorder for almost all realizations of the patterns.

The general theory of inhomogeneous mean-field systems of Ref. [18] provides a variational expression for the (almost sure) limiting free energy density of the Hopfield model H fg N;p (S) = \Gamma 1 2N N X i;j=1 p X =1 i j S i S j for Ising spins S i and p random patterns = ( 1 ; 2 ; \Delta \Delta \Delta ; N ) under the assumption that lim N!1 N \Gamma1 N X i=1 ffi i = ; i = ( 1 i ; 2 i ; \Delta \Delta \Delta ; p i ) exists (almost surely) in the space of probability measure over p copies of f\Gamma1; 1g. Including an "external field" term \Gamma P p =1 h P N i=1 i S i , we give a number of general properties of the free-energy density and compute it for a): p = 2 in general, and b): p arbitrary when is uniform and at most the two components h 1 and h 2 are non-zero, obtaining the (almost sure) formula f(fi; h) = 1 2 f cw (fi; h 1 + h 2 ) + 1 2 f cw (fi; h 1 \Gamma h 2 ) for the free energy, where f cw deno...