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Continuous Spin MeanField models: Limiting kernels and Gibbs Properties of local transforms
, 2008
"... We extend the notion of Gibbsianness for meanfield systems to the setup of general (possibly continuous) local state spaces. We investigate the Gibbs properties of systems arising from an initial meanfield Gibbs measure by application of given local transition kernels. This generalizes previous c ..."
Abstract

Cited by 7 (4 self)
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We extend the notion of Gibbsianness for meanfield systems to the setup of general (possibly continuous) local state spaces. We investigate the Gibbs properties of systems arising from an initial meanfield Gibbs measure by application of given local transition kernels. This generalizes previous casestudies made for spins taking finitely many values to the first step in 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 nonGibbs regime and invokes a minimization problem for a ”constrained ratefunction”. (2) A criterion for Gibbsianness of the transformed system for initial LipschitzHamiltonians involving concentration properties of the transition kernels. (3) A continuity estimate for the singlesite conditional distributions of the transformed system. While (2) and (3) have provable latticecounterparts, the characterization of (1) is stronger in meanfield. As applications we show shorttime Gibbsianness of rotator meanfield models on the (q − 1)dimensional sphere under diffusive timeevolution and the preservation of Gibbsianness under local coarsegraining of the initial local spin space.