Derrida's Generalized Random Energy models 2: models with continuous hierarchies

Abstract 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.

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