94-117 Anton Bovier, Veronique Gayrard, Pierre Picco

Gibbs states of the Hopfield model with extensively many patterns (231K, PS) May 2, 94

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Abstract. We consider the Hopfield model with $M(N)=\alpha N$ patterns, where $N$ is the number of neurons. We show that, if $\alpha$ is sufficiently smalland the temperature s 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 [BGP1]. The key new ingredient is a self averaging result on the free energy functional. This result has consderable additional interest and some consequences are discussed. A similar result for the free energy of the Sherrington-Kirkpatrick model is also given.

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