Self-averaging in a class of generalized Hopfield models

Abstract

We prove the almost sure convergence to zero of the fluctuations of the free energy, resp. local free energies, in a class of disordered mean-field spin systems that generalize the Hopfield model in two ways: 1. multi-spin interactions are permitted and 2. the random variables #xi#_i"#mu# describing the 'patterns' can have arbitrary distributions with mean zero and finite 4+#epsilon#-th moments. The number of patterns, M, is allowed to be an arbitrary multiple of the systemsize. This generalizes a previous result of Bovier, Gayrard, and Picco [BGP3] for the standard Hopfield model, and improves a result of Feng and Tirozzi [FT] that required M to be a finite constant. Note that the convergence of the mean of the free energy is not proven. (orig.)Available from TIB Hannover: RR 5549(105) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman