GIBBS-STATES OF THE HOPFIELD MODEL IN THE REGIME OF PERFECT MEMORY

We study the thermodynamic properties of the Hopfield model of an autoassociative memory. If N denotes the number of neurons and M(N) the number of stored patterns, we prove the following results: If M M/N down arrow 0 as N up arrow infinity, then there exists an infinite number of infinite volume Gibbs measures for all temperatures T < 1 concentrated on spin configurations that have overlap with exactly one specific pattern. Moreover, the measures induced on the overlap parameters are Dirac measures concentrated on a single point and the Gibbs measures on spin configurations are products of Bernoulli measures. If M/N --> alpha, as N up arrow infinity for alpha small enough, we show that for temperatures T smaller than some T(alpha) < 1, the induced measures can have support only on a disjoint union of balls around the previous points, but we cannot construct the infinite volume measures through convergent sequences of measures.