Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Fluctuations of the free energy in the high temperature Hopfield model

Auteur(s):

Code(s) de Classification MSC:

Résumé: We consider the Hopfield model of size $N$ and with $p \sim tN$ patterns, in the whole high temperature (paramagnetic) region. Our result is that the partition function has log-normal fluctuations. It is obtained by extending to the present model the method of the interpolating Brownian Motions used in [10] for the Sherrington-Kirkpatrick model. We view the load $t$ of the memory as a dynamical parameter, making the partition function a nice stochastic process. Then we write some semi-martingale decomposition for the logarithm of the partition function, and we prove that all the terms in this decomposition converge. In particular, the martingale term converges to a Gaussian martingale.

Mots Clés: Hopfield Model ; spin glass ; fluctuations ; martingales

Date: 2003-11-04

Prépublication numéro: PMA-859