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 third of a series of three
papers in which we present a rigorous analysis of Derrida's
Generalized Random Energy Models (GREM). Here we study the general
case of models with a ``continuum of hierarchies''. We prove the
convergence of the free energy and give explicate formulas for the
free energy and the two-replica distribution function. Then we
introduce the empirical distance distribution to describe
effectively the Gibbs measures. We show that its limit is uniquely
determined via the Ghirlanda-Guerra identities up to the mean of
the replica distribution function.
Finally, we show that suitable
discretizations of the limiting random measure can be described by
the same objects in suitably constructed GREM's.

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

**Date:** 2002-05-17

**Prépublication numéro:** *PMA-729*

**Pdf file : **PMA-729.pdf