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Cooperation with: I. Kurkova (Université Paris VI, France)
Supported by: DFG: Dutch-German Bilateral Research Group ``Mathematics of random spatial models from physics and biology''
Description: The investigation of the Generalized Random Energy Models (GREM) described in last year's report ([1], [2]) has been continued this year in [3]. Our aim was to give a more explicit description of the limiting Gibbs measure of these models. To this end we introduced the notion of a flow of probability measures and its associated genealogy. In fact, embedding the Gibbs measure of spin systems together with all its coarse graining in the unit interval, one obtains naturally such a flow of measures, time playing the role of the scale of the coarse graining. It turns out that the associated genealogical structure is precisely equivalent to the multi-overlap structure of the original Gibbs measures. Based on the results of [2] we could then show this flow converges (in the sense that its genealogy converges) to a flow of probability measures associated to a particular continuous-state branching process, introduced by J. Neveu (incidently the same process that was considered in a recent paper by K. Fleischmann and A. Sturm in another project of our group this year). This yields finally a very satisfactory probabilistic and geometric description of Gibbs measures of disordered models and highlights an intimate connection between branching processes and continuous-state branching process.
References:
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