Derrida’s Generalized Random Energy models 2: models with continuous hierarchies
نویسندگان
چکیده
This is the second 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 explicit formulas for the free energy and the two-replica distribution function in thermodynamical limit. 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 GREMs. 2004 Elsevier SAS. All rights reserved. Résumé Cet article est le deuxième d’une série de trois articles où nous présentons l’analyse de Generalized Random Energy Models (GREM) de Derrida. Nous étudions ici le cas général des modèles ayant un “continuum de hierarchies”. Nous prouvons la convergence de l’énergie libre et nous obtenons des formules explicites pour l’énergie libre et la distribution de la distance entre deux répliques dans la limite thermodynamique. Puis, nous introduisons la distribution des distances empiriques pour donner une descrpition complète de la mesure de Gibbs. Nous montrons que sa limite est entièrement déterminée par les identités de Ghirlanda–Guerra sachant l’espérance de la distance entre deux répliques. Finalement, nous montrons que les discrétisations de la mesure aléatoire limite sont définies par les mêmes objets dans les GREMs appropriés. 2004 Elsevier SAS. All rights reserved. MSC: 82B44; 60G70; 60K35
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