Rigorous results on the thermodynamics of the dilute Hopfield model

Abstract

We study the Hopfield model of an autoassociative memory on a random graph on N vertices where the probability of two vertices being joined by a link is p(N). Assuming that p(N) goes to zero more slowly than O(1/N), we prove the following results: 1. If the number of stored patterns, m(N), is small enough such that m(N)/(Np(N)) #arrow down# 0, as N #arrow up# #infinity#, then the free energy of this model converges, upon proper rescaling, to that of the standard Curie-Weiss model, for almost all choices of the random graph and the random patterns. 2. If in addition m(N) #<=#N/ln2, we prove that there exists, for T #<=# 1, a Gibbs measure associated to each original pattern, whereas for higher temperatures the Gibbs measure is unique. The basic technical result in the proofs is an uniform bound on the difference between the Hamiltonian on a random graph and its mean value. (orig.)Available from TIB Hannover: RR 5549(19)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman