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Unité Mixte de Recherche du C.N.R.S. N°7534
Abstract : We consider a class of disordered mean-field spin systems that generalize the Hopfield model with many patterns in two ways: (i) General multi-spin interactions are permitted and (ii) the disorder variables have arbitrary distributions with finite exponential moments. We prove that for all models in this class the high temperature normalized partition function fluctuates according to (essentially) the same log-normal distribution. We also give an analogue statement concerning the fluctuations of the joint distribution of the overlaps of any number of replicas. The key ingredient in the proof of these results is an asymptotic expansion of the Laplace's integralthat we perform up to the $1/N$-term.
Laplace's method and high temperature generalized Hopfield models
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