Much ado about Derrida's GREM
- Bovier, Anton
- Kurkova, Irina
2010 Mathematics Subject Classification
- Gaussian processes, spin-glasses, Generalised random energy model, Poisson point processes, branching processes, coalescence
We provide a detailed analysis of Derrida's Generalised Random Energy Model (GREM). In particular, we describe its limiting Gibbs measure in terms Ruelle's Poisson cascades. Next we introduce and analyse a more general class of Continuous Random Energy Models (CREMs) which differs from the well-known class of Sherrington-Kirkpatrick models only in the choice of distance on the space of spin configurations : the Hamming distance defines the later class while the ultrametric distance corresponds to the former one. We express explicitly the geometry of its limiting Gibbs measure in terms of genealogies of Neveu's Continuous State branching Process via an appropriate time change. We also identify the distances between replicas under the limiting CREM's Gibbs measure with those between integers of Bolthausen-Sznitman coalescent under the same time change.