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

### Temperature dependence of the Gibbs state in the Random Energy Model

Auteur(s):

Code(s) de Classification MSC:

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

Résumé: We consider the problem of temperature dependence of the Gibbs states in two spin-glass models: Derrida's Random Energy Model and its analogue, where the random variables in the Hamiltonian are replaced by independent standard Brownian motions. For both of them we compute in the thermodynamic limit the overlap distribution $\sum_{i=1}^{N}\s_i\s'_i/N\in [0,1]$ of two spin configurations $\s$, $\s'$ under the product of two Gibbs measures, which are taken at temperatures $T, T'$ respectively. If $T\ne T'$ are fixed, then at low temperature phase the results are different for these models: for the first one this distribution is $D_0\delta_0+D_1\delta_2$, with random weights $D_0$, $D_1$, while for the second one it is $\delta_0$. We compute consequently the overlap distribution for the second model whenever $T-T'\to 0$ at different speeds as $N\to \infty$.

Mots Clés: Gaussian processes ; spin-glasses ; random energy model ; overlap ; Poisson point processes

Date: 2002-05-30

Prépublication numéro: PMA-734

Pdf file : PMA-734.pdf