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

TL;DR

This paper analyzes the conditions under which the fuzzy Potts model with Kac interaction on a torus exhibits Gibbsian or non-Gibbsian behavior, establishing sharp thresholds for phase transitions using variational and large deviation techniques.

Contribution

It introduces a variational approach to identify sharp Gibbs-non-Gibbs transition thresholds in the fuzzy Kac-Potts model with unequal class sizes.

Findings

Sharp thresholds for Gibbsian behavior are established.

Large deviation principles for color profiles are proved.

Monotonicity arguments support the main results.

Abstract

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 Fernandez, den Hollander and Mart{\i}nez 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. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments.