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

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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.

Article information

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

First available in Project Euclid: 26 June 2012

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Zentralblatt MATH identifier

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]

Non-Gibbsianness Ising models Tree graphs Cayley tree Glauber dynamics


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.

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