2008/06/04 by Christof Külske, C. Kuelske, Alex Akwasi Opoku +1
Mathematics · Physics and Astronomy · #Gibbs measure #Lipschitz continuity #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematics #Mean field theory #Physics #Quantum mechanics #Spins #Statistical physics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:60K35 #msc:82B20 #msc:82B26
paper · pdf · doi:10.1063/1.3021285
32 pages
arxiv created 2008/06/04 · openalex publication_date 2008/12/01 · arxiv updated 2009/12/01 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/05
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.