## Bernoulli

• Bernoulli
• Volume 25, Number 3 (2019), 2051-2074.

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

#### 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 in (Sharp thresholds for Gibbs-non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction (2017)). 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 (containing precisely two spin values). 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.

#### Article information

Source
Bernoulli, Volume 25, Number 3 (2019), 2051-2074.

Dates
Received: August 2017
Revised: February 2018
First available in Project Euclid: 12 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1560326437

Digital Object Identifier
doi:10.3150/18-BEJ1045

Mathematical Reviews number (MathSciNet)
MR3961240

Zentralblatt MATH identifier
07066249

#### Citation

Henning, Florian; Kraaij, Richard C.; Külske, Christof. Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction: Closing the Ising gap. Bernoulli 25 (2019), no. 3, 2051--2074. doi:10.3150/18-BEJ1045. https://projecteuclid.org/euclid.bj/1560326437

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