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

TL;DR

This paper analyzes the ergodic regime of analogical Hopfield neural networks using a real replica approach, focusing on the high storage limit and the behavior of the partition function's moments.

Contribution

It introduces a real replica method for a generalized Hopfield model with Gaussian patterns and studies the infinite volume behavior of the partition function's moments.

Findings

Identifies a parameter region with self-averaging free energy density.

Calculates corrections to annealed approximations of thermodynamic quantities.

Discusses fluctuations of overlaps as order parameters.

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 accordingly to a standard unit Gaussian. We consider the high storage case, when the number of patterns is linearly diverging 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 is self-averaging 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.