Continuous Spin Mean-Field models: Limiting kernels and Gibbs Properties of local transforms

TL;DR

This paper generalizes the concept of Gibbsianness for mean-field models with continuous state spaces, providing formulas, criteria, and estimates for the Gibbs properties of transformed systems, with applications to rotator models and coarse-graining.

Contribution

It introduces a comprehensive framework for analyzing Gibbsianness in continuous mean-field systems, extending previous finite-spin results to more general settings.

Findings

Derived a formula for limiting conditional probabilities in transformed systems.

Established criteria for Gibbsianness based on transition kernel properties.

Proved short-time Gibbsianness for rotator mean-field models under diffusion.

Abstract

We extend the notion of Gibbsianness for mean-field systems to the set-up 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 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 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. (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.