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We investigate the Gibbs properties of the fuzzy Potts model on thed-dimensional torus with Kac interaction. We use a variational approach forprofiles inspired by that of Fernandez, den Hollander and Mart{\i}nez for theirstudy of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on thetorus. As our main result, we show that the mean-field thresholds dividingGibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model withclass size unequal two. On the way to this result we prove a large deviationprinciple for color profiles with diluted total mass densities and usemonotocity arguments.

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