Continuous spin mean-field models: Limiting kernels and Gibbs properties of local transformsKulske, C. & Opoku, A. A., Dec-2008, In : Journal of Mathematical Physics. 49, 12, 31 p., 125215.
Research output: Contribution to journal › Article › Academic › peer-review
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.
|Number of pages||31|
|Journal||Journal of Mathematical Physics|
|Publication status||Published - Dec-2008|
- lattice theory, phase transformations, RENORMALIZATION-GROUP TRANSFORMATIONS, GIBBSIANNESS, REGULARITY, DIFFUSION, RECOVERY, SYMMETRY