Gibbs-non-Gibbs properties for evolving Ising models on trees
Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 3, pp. 774-791.

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.

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.

DOI: 10.1214/11-AIHP421
Classification: 82C20,  82B20,  60K35
Keywords: non-gibbsianness, Ising models, tree graphs, Cayley tree, Glauber dynamics
@article{AIHPB_2012__48_3_774_0,
     author = {van Enter, Aernout C. D. and Ermolaev, Victor N. and Iacobelli, Giulio and K\"ulske, Christof},
     title = {Gibbs-non-Gibbs properties for evolving {Ising} models on trees},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {774--791},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {3},
     year = {2012},
     doi = {10.1214/11-AIHP421},
     zbl = {1255.82037},
     mrnumber = {2976563},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/11-AIHP421/}
}
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
DA  - 2012///
SP  - 774
EP  - 791
VL  - 48
IS  - 3
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/11-AIHP421/
UR  - https://zbmath.org/?q=an%3A1255.82037
UR  - https://www.ams.org/mathscinet-getitem?mr=2976563
UR  - https://doi.org/10.1214/11-AIHP421
DO  - 10.1214/11-AIHP421
LA  - en
ID  - AIHPB_2012__48_3_774_0
ER  - 
%0 Journal Article
%A van Enter, Aernout C. D.
%A Ermolaev, Victor N.
%A Iacobelli, Giulio
%A Külske, Christof
%T Gibbs-non-Gibbs properties for evolving Ising models on trees
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2012
%P 774-791
%V 48
%N 3
%I Gauthier-Villars
%U https://doi.org/10.1214/11-AIHP421
%R 10.1214/11-AIHP421
%G en
%F AIHPB_2012__48_3_774_0
van Enter, Aernout C. D.; Ermolaev, Victor N.; Iacobelli, Giulio; Külske, Christof. Gibbs-non-Gibbs properties for evolving Ising models on trees. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 3, pp. 774-791. doi : 10.1214/11-AIHP421. http://www.numdam.org/articles/10.1214/11-AIHP421/

[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. | MR | Zbl

[2] D. Dereudre and S. Rœlly. Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions. J. Statist. Phys. 121 (2005) 511-551. | MR | Zbl

[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. | MR | Zbl

[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. | MR

[5] H. O. Georgii. Gibbs Measures and Phase Transitions. de Gruyter, Berlin, 1988. ISBN 0-89925-462-4. | MR | Zbl

[6] O. Häggström. Almost sure quasilocality fails for the random-cluster model on a tree. J. Statist. Phys. 84 (1996) 1351-1361. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[10] C. Külske and F. Redig. Loss without recovery of Gibbsianness during diffusion of continuous spins. Probab. Theory Related Fields 135 (2006) 428-456. | MR | Zbl

[11] A. Le Ny. Fractal failure of quasilocality for a majority rule transformation on a tree. Lett. Math. Phys. 54 (2000) 11-24. | MR | Zbl

[12] A. Le Ny and F. Redig. Short time conservation of Gibbsianness under local stochastic evolutions. J. Statist. Phys. 109 (2002) 1073-1090. | MR | Zbl

[13] A. A. Opoku. On Gibbs measures of transforms of lattice and mean-field systems. Ph.D. thesis, Rijksuniversiteit Groningen, 2009. | Zbl

[14] R. Pemantle and J. Steif. Robust phase tramsitions for Heisenberg and other models on general trees. Ann. Probab. 27 (1999) 876-912. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

Cited by Sources: