 94117 Anton Bovier, Veronique Gayrard, Pierre Picco
 Gibbs states of the Hopfield model with extensively many patterns
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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 SherringtonKirkpatrick model is also given.
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