Continuous spin mean-field models: Limiting kernels and Gibbs properties of local transforms

Christof Kulske*, Alex A. Opoku

*Bijbehorende auteur voor dit werk

    Onderzoeksoutput: ArticleAcademicpeer review

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    We extend the notion of Gibbsianness for mean-field systems to the setup of general (possibly continuous) local state spaces. We investigate the Gibbs properties of systems arising from an initial mean-field Gibbs measure by application of given local transition kernels. This generalizes previous case studies made for spins taking finitely many values to the first step in the direction to a general theory containing the following parts: (1) A formula for the limiting conditional probability distributions of the transformed system (it holds both in the Gibbs and in the non-Gibbs regime and invokes a minimization problem for a "constrained rate function"), (2) a criterion for Gibbsianness of the transformed system for initial Lipschitz-Hamiltonians involving concentration properties of the transition kernels, and (3) a continuity estimate for the single-site conditional distributions of the transformed system. While (2) and (3) have provable lattice counterparts, the characterization of (1) is stronger in mean field. As applications we show short-time Gibbsianness of rotator mean-field models on the (q-1)-dimensional sphere under diffusive time evolution and the preservation of Gibbsianness under local coarse graining of the initial local spin space.

    Originele taal-2English
    Aantal pagina's31
    TijdschriftJournal of Mathematical Physics
    Nummer van het tijdschrift12
    StatusPublished - dec.-2008

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