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:**

- 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*