Variational Description of Gibbs-Non-Gibbs Dynamical Transitions for Spin-Flip Systems with a Kac-Type Interaction

Abstract

We continue our study of Gibbs-non-Gibbs dynamical transitions. In the present paper we consider a system of Ising spins on a large discrete torus with a Kac-type interaction subject to an independent spin-flip dynamics (infinite-temperature Glauber dynamics). We show that, in accordance with the program outlined in [11], in the thermodynamic limit Gibbs-non-Gibbs dynamical transitions are equivalent to bifurcations in the set of global minima of the large-deviation rate function for the trajectories of the empirical density conditional on their endpoint. More precisely, the time-evolved measure is non-Gibbs if and only if this set is not a singleton for some value of the endpoint. A partial description of the possible scenarios of bifurcation is given, leading to a characterization of passages from Gibbs to non-Gibbs and vice versa, with sharp transition times.Our analysis provides a conceptual step-up from our earlier work on Gibbs-non-Gibbs dynamical transitions for the Curie-Weiss model, where the mean-field interaction allowed us to focus on trajectories of the empirical magnetization rather than the empirical density.MSC 2010. 60F10, 60K35, 82C22, 82C27.

Sharp thresholds for Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction

“…The first rigorous result relating Gibbs properties of a KM to that of a meanfield model was obtained in [18] in the case of independent time evolutions from an initial Kac-Ising model. The relation between a spatial model and a meanfield model was set up as follows.…”

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We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernández, den Hollander and Martínez [18] for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model with class size unequal two.

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Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction: Closing the Ising gap

“…The Gibbs property is an important regularity property of a large, or infinite system which comes in various versions, according to the setting considered. It can be formulated for lattice models in the infinite volume [EFS93,Geo11], for systems of point particles in euclidean space [Rue99] for mean-field systems [KLN07], or for Kac-systems [FdHM14].…”

“…Stochastic time-evolutions, motivated from physics, provide another very interesting type of such transformations, cf. [KLN07,EK10,FdHM13,FdHM14,dHRvZ15,JK16,JK17a].…”

“…The Kac fuzzy Potts model is defined as the image of the Kac-version of Potts model, under the deterministic local transformation which distinguishes the possible spin values of the Potts local spin space {1, 2, ..., q} only according to a fixed local partitioning into r subclasses. The Kac-Potts model itself is formulated on a d-dimensional torus, a setup similar to that of [FdHM14] in their study of the dynamical Kac-Ising model. We are now able to completely answer the question:…”