About the ergodic regime in the analogical Hopfield neural networks: Moments of the partition function

Abstract

In this paper we introduce and exploit the real replica approach for a minimal generalization of the Hopfield model by assuming the learned patterns to be distributed according to a standard unit Gaussian. We consider the high storage case, when the number of patterns linearly diverges with the number of neurons. We study the infinite volume behavior of the normalized momenta of the partition function. We find a region in the parameter space where the free energy density in the infinite volume limit self-averages around its annealed approximation, as well as the entropy and the internal energy density. Moreover, we evaluate the corrections to their extensive counterparts with respect to their annealed expressions. The fluctuations of properly introduced overlaps, which act as order parameters, are also discussed.