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

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 meanfield 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.

Gibbs-Non-Gibbs Transitions via Large Deviations: Computable Examples

“…Therefore, if we have at least two optimal trajectories conditioned to arrive at a certain magnetization m * at time T > 0, and these trajectories can be selected by approximating the magnetization appropriately, then we have an essential discontinuity at m = m * of the conditional distribution m → μ F n (t)(dx 1 |m) as a function of the magnetization m = (1/n) n i=2 x i . Such a discontinuity is referred to as non-Gibbsianness in the meanfield context, see [5,7] for more details.…”

“…Multiple histories were then shown to lead to jumps in conditional probabilities indicating non-Gibbsian behavior in the mean-field setting, see e.g. [5,7,14]. These special conditionings leading to multiple histories are the analogue of "bad configurations" (essential points of discontinuity of conditional probabilities of the measure at time t ) in the (lattice) Gibbs-non-Gibbs transition scenario.…”

A Large-Deviation View on Dynamical Gibbs-Non-Gibbs Transitions

“…Bad configurations can be characterized explicitly (with the interesting effect that non-neutral bad configurations can arise below a certain critical temperature). For further developments on mean-field results see also [16], [10].…”

Variational Description of Gibbs-non-Gibbs Dynamical Transitions for the Curie-Weiss Model

“…A particularly fruitful research direction was initiated by Külske and Le Ny [9], who showed that Gibbs-non-Gibbs transitions can also be defined naturally for mean-field models, such as the Curie-Weiss model. Precise results are available for the latter, including sharpness of the transition times and an explicit characterization of the conditional magnetizations leading to non-Gibbsianness (Külske and Opoku [11], Ermolaev and Külske [7]). In particular, the work in [7] shows that in the meanfield setting Gibbs-non-Gibbs transitions occur for all initial temperatures below criticality, both for cooling dynamics and for heating dynamics.…”