An almost sure large deviation principle for the Hopfield model

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 lim sup $M(N) /N =0$. In this case, the rate function is independent of the disorder for almost all realizations of the patterns.