A central limit theorem for the overlap in the Hopfield model

Abstract

We consider the Hopfield model with n neurons and an increasing number $p = p(n)$ of randomly chosen patterns. Under the condition $(p^3 \log p)/n \to 0$, we prove for every fixed choice of overlap parameters a central limit theorem as $n \to \infty$, which holds for almost all realizations of the random patterns. In the special case where the temperature is above the critical one and there is no external magnetic field, the condition $(p^2 \log p)/n \to 0$ suffices. As in the case of a finite number of patterns, the central limit theorem requires a centering which depends on the random patterns.