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

By 13 - Marseille (France). Centre de Physique Theorique) P. (Centre National de la Recherche Scientifique (CNRS) Picco, V. Gayrard, Berlin (Germany)) A. (Institut fuer Angewandte Analysis und Stochastik (IAAS) Bovier and Berlin (Germany) Institut fuer Angewandte Analysis und Stochastik (IAAS)

Abstract

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/N #arrow down# 0 as N #arrow up# #infinity#, then there extists 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. If M/N #-># #alpha#, as N #arrow up# #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. (orig.)SIGLEAvailable from TIB Hannover: RR 5549(79)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Topics: 12C - Applied mathematics, HOPFIELD MODEL: M, THERMODYNAMIC PROPERTIES, GIBBS MEASURES, DIRAC MEASURES, LAPLACE METHOD, DENSITY, GLOBAL MINIMA, RANDOM MATRIX
Year: 1994
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Provided by: OpenGrey Repository
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