Free energy of the Hopfield model

Abstract

We identify the limit of the free energy of the Hopfield model with independent symmetric Bernoulli patterns, for every temperature and loading ratio. A Hubbard-Stratonovich transformation reduces the problem to that of identifying the limit free energy of a bipartite spin glass with $\pm 1$ and spherical spins. One key difficulty is that the patterns, which become the coupling coefficients in the bipartite model, are Bernoulli instead of Gaussian, and the limit free energy is sensitive to the law of these coefficients. We show that the model is in fact equivalent to one involving Gaussian patterns and also subject to a finite number of Bernoulli random fields. We then solve this model by adapting the techniques recently introduced to identify the limit free energies of spin glasses with either $\pm 1$ or spherical spins.