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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60K35 Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
- 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.)
- 60G42 Martingales with discrete parameter

**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*