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

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

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 3, page 774-791
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

van Enter, Aernout C. D., et al. "Gibbs–non-Gibbs properties for evolving Ising models on trees." Annales de l'I.H.P. Probabilités et statistiques 48.3 (2012): 774-791. <http://eudml.org/doc/272050>.

@article{vanEnter2012,
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.},
author = {van Enter, Aernout C. D., Ermolaev, Victor N., Iacobelli, Giulio, Külske, Christof},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {non-gibbsianness; Ising models; tree graphs; Cayley tree; Glauber dynamics; non-Gibbsianness},
language = {eng},
number = {3},
pages = {774-791},
publisher = {Gauthier-Villars},
title = {Gibbs–non-Gibbs properties for evolving Ising models on trees},
url = {http://eudml.org/doc/272050},
volume = {48},
year = {2012},
}

TY - JOUR
AU - van Enter, Aernout C. D.
AU - Ermolaev, Victor N.
AU - Iacobelli, Giulio
AU - Külske, Christof
TI - Gibbs–non-Gibbs properties for evolving Ising models on trees
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 3
SP - 774
EP - 791
AB - 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.
LA - eng
KW - non-gibbsianness; Ising models; tree graphs; Cayley tree; Glauber dynamics; non-Gibbsianness
UR - http://eudml.org/doc/272050
ER -

References

top
  1. [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. Zbl1081.82515MR1325591
  2. [2] D. Dereudre and S. Rœlly. Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions. J. Statist. Phys.121 (2005) 511–551. Zbl1127.82039MR2185338
  3. [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. Zbl1208.82042MR2739308
  4. [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. Zbl05723808MR2581896
  5. [5] H. O. Georgii. Gibbs Measures and Phase Transitions. de Gruyter, Berlin, 1988. ISBN 0-89925-462-4. Zbl1225.60001MR956646
  6. [6] O. Häggström. Almost sure quasilocality fails for the random-cluster model on a tree. J. Statist. Phys.84 (1996) 1351–1361. Zbl1081.82520MR1412082
  7. [7] O. Häggström and C. Külske. Gibbs properties of the fuzzy Potts model on trees and in mean field. Markov Process. Related Fields10 (2004) 477–506. Zbl1210.82023MR2097868
  8. [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. Zbl0848.60090MR1391195
  9. [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. Zbl1190.60096MR2438808
  10. [10] C. Külske and F. Redig. Loss without recovery of Gibbsianness during diffusion of continuous spins. Probab. Theory Related Fields135 (2006) 428–456. Zbl1095.60027MR2240694
  11. [11] A. Le Ny. Fractal failure of quasilocality for a majority rule transformation on a tree. Lett. Math. Phys.54 (2000) 11–24. Zbl0989.82013MR1846719
  12. [12] A. Le Ny and F. Redig. Short time conservation of Gibbsianness under local stochastic evolutions. J. Statist. Phys.109 (2002) 1073–1090. Zbl1015.60097MR1938286
  13. [13] A. A. Opoku. On Gibbs measures of transforms of lattice and mean-field systems. Ph.D. thesis, Rijksuniversiteit Groningen, 2009. Zbl1192.82004
  14. [14] R. Pemantle and J. Steif. Robust phase tramsitions for Heisenberg and other models on general trees. Ann. Probab.27 (1999) 876–912. Zbl0981.60096MR1698979
  15. [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. Zbl1188.82047MR2601426
  16. [16] A. C. D. van Enter and W. M. Ruszel. Loss and recovery of Gibbsianness for X Y spins in small external fields. J. Math. Phys. 49 (2008) 125208. Zbl1159.81345MR2484339
  17. [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. Zbl1173.82007MR2519348
  18. [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. Zbl0990.82018MR1889994
  19. [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. Zbl1221.82046MR2791053
  20. [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. Zbl1101.82314MR1241537
  21. [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. Zbl1200.82015MR2643565

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.