Laplace's method and high temperature generalized Hopfield models
Trashorras, José (2006), Laplace's method and high temperature generalized Hopfield models, Markov Processes and Related Fields, 12, 3, p. 583-626
TypeArticle accepté pour publication ou publié
Journal nameMarkov Processes and Related Fields
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Abstract (EN)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 distri- butions 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 integral that we perform up to the 1/N-term.
Subjects / KeywordsHopﬁeld models; Laplace’s method; Large Deviations; Fluctuations; Martingales
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