Gibbs-non-Gibbs transition in the fuzzy Potts models with a Kac-type interaction: Closing the Ising gap

TL;DR

This paper investigates the Gibbs properties of fuzzy Potts models with Kac interactions, extending mean-field bounds to include Ising classes and closing the previously open Ising-gap through analytical methods.

Contribution

It extends the sharpness of mean-field bounds to all fuzzy transformations, including Ising classes, and proves uniqueness of minimizing profiles for non-homogeneous problems.

Findings

Extended Gibbs property analysis to all fuzzy transformations

Closed the Ising-gap in the Gibbs property characterization

Proved uniqueness of minimizing profiles in non-homogeneous cases

Abstract

We complete the investigation of the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction which was started by Jahnel and one of the authors. As our main result of the present paper, we extend the previous sharpness result of mean-field bounds to cover all possible cases of fuzzy transformations, allowing also for the occurrence of Ising classes. The closing of this previously left open Ising-gap involves an analytical argument showing uniqueness of minimizing profiles for certain non-homogeneous conditional variational problems.