Derrida's Generalized Random Energy models 4: Continuous state branching and coalescents
- Bovier, Anton
- Kurkova, Irina
2010 Mathematics Subject Classification
- 82B44 60G70 60K35
- Gaussian processes, generalized random energy model, continuous state branching process, subordinators, coalescent processes, genealogy, Ghirlanda-Guerra identities
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.