Gibbs states of the Hopfield model in the regime of perfect memory

Summary We study the thermodynamic properties of the Hopfield model of an autoassociative memory. IfN denotes the number of neurons andM (N) the number of stored patterns, we prove the following results: IfM/N 0 asN , then there exists an infinite number of infinite volume Gibbs measures for all temperaturesT<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. IfM/N , asN for small enough, we show that for temperaturesT smaller than someT()<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.Work partially supported by the Commission of the European Communities under contract No. SC1-CT91-0695