Anton Bovier, Veronique Gayrard, Pierre Picco
Gibbs states of the Hopfield model with extensively many patterns
(231K, PS)

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.