Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Gibbs–non-Gibbs properties for evolving Ising models on trees

Aernout C. D. van Enter, Victor N. Ermolaev, Giulio Iacobelli, and Christof Külske

Full-text: Open access

Abstract

In this paper we study homogeneous Gibbs measures on a Cayley tree, subjected to an infinite-temperature Glauber evolution, and consider their (non-)Gibbsian properties. We show that the intermediate Gibbs state (which in zero field is the free-boundary-condition Gibbs state) behaves differently from the plus and the minus state. E.g. at large times, all configurations are bad for the intermediate state, whereas the plus configuration never is bad for the plus state. Moreover, we show that for each state there are two transitions. For the intermediate state there is a transition from a Gibbsian regime to a non-Gibbsian regime where some, but not all configurations are bad, and a second one to a regime where all configurations are bad.

For the plus and minus state, the two transitions are from a Gibbsian regime to a non-Gibbsian one and then back to a Gibbsian regime again.

Résumé

Dans cet article, nous étudions les mesures homogènes de Gibbs sur un arbre de Cayley soumises à une évolution de Glauber à une température infinie, et nous considérons leurs propriétés dites « non Gibbsiennes ». Nous montrons que l’ état de Gibbs intermédiaire (c’est à dire pour un champ magnétique nul l’état de Gibbs correspondant à la condition au bord libre) se comporte différemment des états de Gibbs « plus » et « moins ». Par exemple, lorsque le temps est assez grand, toutes les configurations sont mauvaises pour l’état intermédiaire, tandis que la configuration « plus » n’est jamais mauvaise pour l’état « plus ». De plus nous montrons que, pour chaque état, il y a deux transitions. Pour l’état intermédiaire il y a une première transition d’un régime Gibbsien à un régime non-Gibbsien, où certaines configurations mais pas toutes sont mauvaises. Après cette première transition, il y en a une seconde dans laquelle l’état intermédiaire passe à un régime où toutes les configurations sont mauvaises.

Pour les états « plus » et « moins », il y a également deux transitions : une première d’un régime Gibbsien à un régime non-Gibbsien, et une deuxième d’un régime non-Gibbsien à un régime Gibbsien.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 3 (2012), 774-791.

Dates
First available in Project Euclid: 26 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1340714872

Digital Object Identifier
doi:10.1214/11-AIHP421

Mathematical Reviews number (MathSciNet)
MR2976563

Zentralblatt MATH identifier
1255.82037

Subjects
Primary: 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Non-Gibbsianness Ising models Tree graphs Cayley tree Glauber dynamics

Citation

van Enter, Aernout C. D.; Ermolaev, Victor N.; Iacobelli, Giulio; Külske, Christof. Gibbs–non-Gibbs properties for evolving Ising models on trees. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 3, 774--791. doi:10.1214/11-AIHP421. https://projecteuclid.org/euclid.aihp/1340714872


Export citation

References

  • [1] P. M. Bleher, J. Ruiz and V. A. Zagrebnov. On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Statist. Phys. 79 (1995) 473–482.
  • [2] D. Dereudre and S. Rœlly. Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions. J. Statist. Phys. 121 (2005) 511–551.
  • [3] V. N. Ermolaev and C. Külske. Low-temperature dynamics of the Curie–Weiss model: Periodic orbits, multiple histories and loss of Gibbsianness. J. Statist. Phys. 141 (2010) 727–756.
  • [4] R. Fernández. Gibbsianness and non-Gibbsianness in lattice random fields. In Les Houches Summer School, Session LXXXIII, 2005. Mathematical Statistical Physics, A. Elsevier, Amsterdam, 2006.
  • [5] H. O. Georgii. Gibbs Measures and Phase Transitions. de Gruyter, Berlin, 1988. ISBN 0-89925-462-4.
  • [6] O. Häggström. Almost sure quasilocality fails for the random-cluster model on a tree. J. Statist. Phys. 84 (1996) 1351–1361.
  • [7] O. Häggström and C. Külske. Gibbs properties of the fuzzy Potts model on trees and in mean field. Markov Process. Related Fields 10 (2004) 477–506.
  • [8] D. Ioffe. On the extremality of the disordered state for the Ising model on the Bethe lattice. Lett. Math. Phys. 37 (1996) 137–143.
  • [9] C. Külske and A. A. Opoku. The posterior metric and the goodness of Gibbsianness for transforms of Gibbs measures. Electron. J. Probab. 13 (2008) 1307–1344.
  • [10] C. Külske and F. Redig. Loss without recovery of Gibbsianness during diffusion of continuous spins. Probab. Theory Related Fields 135 (2006) 428–456.
  • [11] A. Le Ny. Fractal failure of quasilocality for a majority rule transformation on a tree. Lett. Math. Phys. 54 (2000) 11–24.
  • [12] A. Le Ny and F. Redig. Short time conservation of Gibbsianness under local stochastic evolutions. J. Statist. Phys. 109 (2002) 1073–1090.
  • [13] A. A. Opoku. On Gibbs measures of transforms of lattice and mean-field systems. Ph.D. thesis, Rijksuniversiteit Groningen, 2009.
  • [14] R. Pemantle and J. Steif. Robust phase tramsitions for Heisenberg and other models on general trees. Ann. Probab. 27 (1999) 876–912.
  • [15] F. Redig, S. Rœlly and W. Ruszel. Short-time Gibbsianness for infinite-dimensional diffusions with space–time interaction. J. Statist. Phys. 138 (2010) 1124–1144.
  • [16] A. C. D. van Enter and W. M. Ruszel. Loss and recovery of Gibbsianness for $XY$ spins in small external fields. J. Math. Phys. 49 (2008) 125208.
  • [17] A. C. D. van Enter and W. M. Ruszel. Gibbsianness versus non-Gibbsianness of time-evolved planar rotor models. Stochastic Processes Appl. 119 (2009) 1866–1888.
  • [18] A. C. D. van Enter, R. Fernández, F. Den Hollander and F. Redig. Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures. Comm. Math. Phys. 226 (2002) 101–130.
  • [19] A. C. D. van Enter, R. Fernández, F. Den Hollander and F. Redig. A large-deviation view on dynamical Gibbs–non-Gibbs transitions. Mosc. Math. J. 10 (2010) 687–711.
  • [20] A. C. D. van Enter, R. Fernández and A. D. Sokal. Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory. J. Statist. Phys. 72 (1993) 879–1167.
  • [21] A. C. D. van Enter, C. Külske, A. A. Opoku and W. M. Ruszel. Gibbs–non-Gibbs properties for $n$-vector lattice and mean-field models. Braz. J. Probab. Stat. 24 (2010) 226–255.