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

Christof Kulske*, Alex A. Opoku

*Corresponding author for this work

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    Abstract

    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.

    Original languageEnglish
    Article number125215
    Number of pages31
    JournalJournal of Mathematical Physics
    Volume49
    Issue number12
    DOIs
    Publication statusPublished - Dec-2008

    Keywords

    • lattice theory
    • phase transformations
    • RENORMALIZATION-GROUP TRANSFORMATIONS
    • GIBBSIANNESS
    • REGULARITY
    • DIFFUSION
    • RECOVERY
    • SYMMETRY

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