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

Florian Henning, Richard C. Kraaij, and Christof Külske

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

Keywords
diluted large deviation principle fuzzy Kac–Potts model Gibbs versus non-Gibbs Kac model large deviation principles Potts model

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


Export citation

References

  • [1] Bogachev, V.I. (2007). Measure Theory. Vol. I, II. Berlin: Springer.
  • [2] den Hollander, F., Redig, F. and van Zuijlen, W. (2015). Gibbs–non-Gibbs dynamical transitions for mean-field interacting Brownian motions. Stochastic Process. Appl. 125 371–400.
  • [3] Ermolaev, V. and Külske, C. (2010). Low-temperature dynamics of the Curie–Weiss model: Periodic orbits, multiple histories, and loss of Gibbsianness. J. Stat. Phys. 141 727–756.
  • [4] Fernández, R., den Hollander, F. and Martínez, J. (2013). Variational description of Gibbs–non-Gibbs dynamical transitions for the Curie–Weiss model. Comm. Math. Phys. 319 703–730.
  • [5] Fernández, R., den Hollander, F. and Martínez, J. (2014). Variational description of Gibbs–non-Gibbs dynamical transitions for spin-flip systems with a Kac-type interaction. J. Stat. Phys. 156 203–220.
  • [6] Georgii, H.-O. (2011). Gibbs Measures and Phase Transitions, 2nd ed. De Gruyter Studies in Mathematics 9. Berlin: de Gruyter.
  • [7] Häggström, O. (2003). Is the fuzzy Potts model Gibbsian? Ann. Inst. Henri Poincaré Probab. Stat. 39 891–917.
  • [8] Häggström, O. and Külske, C. (2004). Gibbs properties of the fuzzy Potts model on trees and in mean field. Markov Process. Related Fields 10 477–506.
  • [9] Jahnel, B. and Külske, C. (2017). Gibbsian representation for point processes via hyperedge potentials. Preprint. Available at arXiv:1707.05991.
  • [10] Jahnel, B. and Külske, C. (2017). Sharp thresholds for Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction. Bernoulli 23 2808–2827.
  • [11] Jahnel, B. and Külske, C. (2017). The Widom–Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality. Ann. Appl. Probab. 27 3845–3892.
  • [12] Klenke, A. (2014). Probability Theory. A Comprehensive Course, 2nd ed. Universitext. London: Springer.
  • [13] Külske, C. (1999). (Non-) Gibbsianness and phase transitions in random lattice spin models. Markov Process. Related Fields 5 357–383.
  • [14] Külske, C. and Le Ny, A. (2007). Spin-flip dynamics of the Curie–Weiss model: Loss of Gibbsianness with possibly broken symmetry. Comm. Math. Phys. 271 431–454.
  • [15] Maes, C. and Vande Velde, K. (1995). The fuzzy Potts model. J. Phys. A 28 4261–4270.
  • [16] Ruelle, D. (1999). Statistical Mechanics: Rigorous Results. River Edge, NJ: World Scientific. London: Imperial College Press.
  • [17] van Enter, A.C.D., Fernández, R. and Sokal, A.D. (1993). Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory. J. Stat. Phys. 72 879–1167.