An Almost Sure Central Limit Theorem for the Hopfield Model

A. Bovier, V. Gayrard

1997, v.3, №2, 151-173

ABSTRACT

We prove a central limit theorem 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 $\lim_{N\uparrow\infty} M(N)/N=0$, without any assumptions on the speed of convergence. The covariance matrix of the limiting gaussian distributions is diagonal and independent of the disorder for almost all realizations of the patterns.

Keywords: Hopfield model,neural networks,central limit theorem,Brascamp-Lieb inequalities