Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Fluctuations of the free energy and overlaps in the high temperature $p$-spin SK and Hopfield models

Auteur(s):

Code(s) de Classification MSC:

• 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.)

Résumé: We study the fluctuations of the free energy and overlaps of $n$ replicas for the $p$-spin Sherrington-Kirkptarick and Hopfield models of spin glasses in the high temperature phase. For the first model we show that at all inverse temperatures $\beta$ smaller than Talagrand's bound $\beta_p$ the free energy on the scale $N^{1-(p-2)/2}$ converges to a Gaussian law with zero mean and variance $\b^4 p!/2$; and that the law of the overlaps $\s\cdot \s'=\sum_{i=1}^{N}\s_i\s'_i$ of $n$ replicas on the scale $\sqrt{N}$ under the product of Gibbs measures is asymptotically the one of $n(n-1)/2$ independent standard Gaussian random variables. For the second model we prove that for all $\beta$ and the load of the memory $t$ with $\beta(1+\sqrt{t})<1$ the law of the overlaps of $n$ replicas on the scale $\sqrt{N}$ under the product of Gibbs measures is asymptotically the one of $n(n-1)/2$ independent Gaussian random variables with zero mean and variance $(1-t\b^2(1-\b)^{-2})^{-1}$.

Mots Clés: Spin glasses ; Sherrington-Kirkpatrick model ; $p$-spin model ; Hopfield model ; overlap ; free energy ; martingales

Date: 2003-11-04

Prépublication numéro: PMA-860