On the Central Limit Theorem for the Overlap in the Hopfield Model

We consider the Hopfield model with N neurons and an increasing number M = M(N) of randomly chosen patterns. Under the condition M2/N → 0, we prove for every fixed choice of overlap parameters a central limit theorem as N → ∞, 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 M3/2 log M ≤ N suffices. As in the case of a finite number of patterns, the central limit theorem requires a centering which depends on the random patterns. In addition, we describe the almost sure asymptotic behavior of the partition function under the condition M3/N → 0.