• Bernoulli
  • Volume 23, Number 4A (2017), 2808-2827.

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

Benedikt Jahnel and Christof Külske

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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 Fernández, den Hollander and Martínez [J. Stat. Phys. 156 (2014) 203–220] 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.

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Bernoulli, Volume 23, Number 4A (2017), 2808-2827.

Received: February 2015
Revised: November 2015
First available in Project Euclid: 9 May 2017

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


Jahnel, Benedikt; Külske, Christof. Sharp thresholds for Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction. Bernoulli 23 (2017), no. 4A, 2808--2827. doi:10.3150/16-BEJ828.

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  • [1] Bálint, A. (2010). Gibbsianness and non-Gibbsianness in divide and color models. Ann. Probab. 38 1609–1638.
  • [2] Bauer, H. (1978). Wahrscheinlichkeitstheorie und Grundzüge der Masstheorie, revised ed. Berlin/New York: Walter de Gruyter.
  • [3] Benois, O., Mourragui, M., Orlandi, E., Saada, E. and Triolo, L. (2012). Quenched large deviations for Glauber evolution with Kac interaction and random field. Markov Process. Related Fields 18 215–268.
  • [4] Bovier, A. and Külske, C. (2005). Coarse-graining techniques for (random) Kac models. In Interacting Stochastic Systems 11–28. Berlin: Springer.
  • [5] Comets, F. (1987). Nucleation for a long range magnetic model. Ann. Inst. Henri Poincaré Probab. Stat. 23 135–178.
  • [6] Comets, F. (1989). Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures. Probab. Theory Related Fields 80 407–432.
  • [7] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Berlin: Springer. Corrected reprint of the second (1998) edition.
  • [8] 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.
  • [9] De Masi, A., Orlandi, E., Presutti, E. and Triolo, L. (1994). Stability of the interface in a model of phase separation. Proc. Roy. Soc. Edinburgh Sect. A 124 1013–1022.
  • [10] De Roeck, W., Maes, C., Netočný, K. and Schütz, M. (2015). Locality and nonlocality of classical restrictions of quantum spin systems with applications to quantum large deviations and entanglement. J. Math. Phys. 56 023301, 30.
  • [11] Eisele, T. and Ellis, R.S. (1983). Symmetry breaking and random waves for magnetic systems on a circle. Z. Wahrsch. Verw. Gebiete 63 297–348.
  • [12] Ellis, R.S. and Wang, K. (1990). Limit theorems for the empirical vector of the Curie–Weiss–Potts model. Stochastic Process. Appl. 35 59–79.
  • [13] Fernández, R. (2006). Gibbsianness and non-Gibbsianness in lattice random fields. In Mathematical Statistical Physics 731–799. Amsterdam: Elsevier B. V.
  • [14] 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.
  • [15] Häggström, O. (2003). Is the fuzzy Potts model Gibbsian? Ann. Inst. Henri Poincaré Probab. Stat. 39 891–917.
  • [16] 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.
  • [17] Jahnel, B., Külske, C., Rudelli, E. and Wegener, J. (2014). Gibbsian and non-Gibbsian properties of the generalized mean-field fuzzy Potts-model. Markov Process. Related Fields 20 601–632.
  • [18] Klenke, A. (2008). Wahrscheinlichkeitstheorie. Berlin/Heidelberg: Springer.
  • [19] Külske, C. (2003). Analogues of non-Gibbsianness in joint measures of disordered mean field models. J. Stat. Phys. 112 1079–1108.
  • [20] 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.
  • [21] Külske, C. and Rozikov, U.A. (2016). Fuzzy transformations of Gibbs measures for the Potts model on a Cayley tree. Random Structures Algorithms. To appear.
  • [22] Le Ny, A. (2008). Gibbsian description of mean-field models. In In and Out of Equilibrium. 2 (V.Sidoravicius and M.E. Vares, eds.). Progress in Probability 60 463–480. Basel: Birkhäuser.
  • [23] Potts, R.B. (1952). Some generalized order–disorder transformations. Proc. Cambridge Philos. Soc. 48 106–109.
  • [24] van Enter, A.C.D. (2012). On the prevalence of non-Gibbsian states in mathematical physics. IAMP News Bulletin 15 15–24.
  • [25] van Enter, A.C.D., Ermolaev, V.N., Iacobelli, G. and Külske, C. (2012). Gibbs–non-Gibbs properties for evolving Ising models on trees. Ann. Inst. Henri Poincaré Probab. Stat. 48 774–791.
  • [26] van Enter, A.C.D., Fernández, R., den Hollander, F. and Redig, F. (2002). Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures. Comm. Math. Phys. 226 101–130.
  • [27] van Enter, A.C.D., Fernández, R., den Hollander, F. and Redig, F. (2010). A large-deviation view on dynamical Gibbs–non-Gibbs transitions. Mosc. Math. J. 10 687–711, 838.
  • [28] 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.