Gibbs states of the Hopfield model with extensively many patterns |
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Authors: | Anton Bovier Véronique Gayrard Pierre Picco |
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Affiliation: | (1) Weierstrass-Institut für Angewandte Analysis und Stochastik, D-10117 Berlin, Germany;(2) Centre de Physique Théorique-CNRS, Luminy, Case 907, F-13288 Marseille Cedex 9, France |
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Abstract: | We consider the Hopfield model withM(N)=N patterns, whereN is the number of neurons. We show that if 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 provlem 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. |
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Keywords: | Hopfield model Gibbs states self-averaging spin glasses |
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