Laplace's Method and High Temperature Generalized Hopfield Models

#### J. Trashorras

2006, v.12, Issue 3, 583-626

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 analogous 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.

Keywords: Hopfield models,Laplace's method,large deviations,fluctuations,martingales