Journal of Statistical Physics, Vol. 79, Nos. 1/2, 1995, p. 395-414.
We consider the Hopfield model with M(N)= r patterns, where N is the number of neurons. We show that if at is sufficiently small and the temperature sufficiently low, then there exist disjoint Gibbs states for each of the stored patterns, almost surely with respect to the distribution of the random patterns. This solves a problem left open in previous work. The key new ingredient is a self-averaging result on the free energy functional. This result has considerable additional interest and some consequences are discussed. A similar result for the free energy of the Sherrington-Kirkpatrick model is also given.