Hopfield models as generalized random mean field models

TL;DR

This paper provides a detailed review and rigorous analysis of the thermodynamics of Hopfield models, focusing on low temperature phases, Gaussian convergence of local fields, and validation of the replica symmetric solution.

Contribution

It offers a comprehensive, self-contained analysis of Hopfield models, including convergence of mean fields, propagation of chaos, and validation of the replica symmetric solution in certain parameter regimes.

Findings

Convergence of local mean fields to Gaussian variables with site-dependent means.

Proof of propagation of chaos and factorization of Gibbs measures.

Validation of the replica symmetric solution in specific regimes.

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.