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 Schrodinger operators, etc.)
- 60G70 Extreme value theory; extremal processes
- 60K35 Interacting random processes; statistical mechanics type models; percolation theory, See also {82B43, 82C43}

**Résumé:** This is the second in a series of three
papers in which we present a full rigorous analysis of Derrida's
Generalized Random Energy Models (GREM). In this paper we still
consider only models with finitely many levels. In this context we
present two ways to prove the convergence of the Gibbs measures
(in a suitable representation). The first is direct and shows
convergence to the probability cascades introduced by Ruelle. The
second approach is indirect and use the so-called Ghirlanda-Guerra
identities, that allow to control the distribution of the Gibbs
measures via recursive identities for their moments.

**Mots Clés:** *Gaussian processes ; generalized random energy model ; spin glasses ; Poisson cascades ; probability cascades ;
Ghirlanda-Guerra identities*

**Date:** 2002-05-17

**Prépublication numéro:** *PMA-728*

**Pdf file : **PMA-728.pdf