Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
| ||

``Probabilités et Modèles Aléatoires''
| ||

**Auteur(s): **

**Code(s) de Classification MSC:**

- 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.)
- 60G70 Extreme value theory; extremal processes
- 60K35 Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Résumé:** In this paper we conclude our analysis of
Derrida's Generalized
Random Energy Models (GREM) by identifying the thermodynamic limit
with a continuous state branching process introduced by Neveu.
Using a construction introduced by Bertoin and Le Gall in terms
of a coherent family of subordinators related to Neveu's branching process,
we show how the Gibbs geometry of
the limiting Gibbs measure is given in terms of this process
via a deterministic time-change. This
construction is fully universal in that all different models (characterized
by the covariance of the underlying Gaussian process) differ only through
that
time change, which in turn is expressed in terms of Parisi's overlap
distribution. The proof uses strongly the Ghirlanda-Guerra identities that
impose the structure of Neveu's process as the only possible asymptotic
random mechanism.

**Mots Clés:** *Gaussian processes ; generalized random energy model ; continuous state branching process ; subordinators ; coalescent processes ;
Ghirlanda-Guerra identities*

**Date:** 2003-06-16

**Prépublication numéro:** *PMA-831*