An almost sure large deviation principle for the Hopfield model

TL;DR

This paper establishes an almost sure large deviation principle for the overlap parameters in the Hopfield model, showing the rate function's independence from disorder under certain conditions.

Contribution

It provides the first large deviation principle for the Hopfield model's overlap parameters with a novel explicit variational rate function.

Findings

Rate function is independent of disorder for most pattern realizations.

Large deviation principle applies when the number of patterns grows slower than the system size.

Explicit variational formula for the rate function is derived.

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 $\limsup M(N)/N=0$. In this case the rate function (or free energy as a function of the overlap parameters) is independent of the disorder for almost all realization of the patterns and given by an explicit variational formula.