Anton Bovier (WIAS), Irina Kurkova (Paris 6)
Derrida's Generalized Random Energy models 4: Continuous state branching and coalescents
(349K, ps)

ABSTRACT.  In this paper we conclude our analysis of 
 Derrida's Generalized 
Random Energy Models (GREM) by identifying the thermodynamic limit 
with a one-parameter family of probability measures related 
to 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 the genealogy 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.