Mixing time for the Ising model : a uniform lower bound for all graphs
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1020-1028.

Dans cet article nous étudions la dynamique de Glauber du modèle d'Ising sur un graphe fini à n sommets. Hayes et Sinclair ont montré que le temps de mélange de cette dynamique est au moins de nlog(n)f(Δ), où Δ est le degré maximum d'un sommet du graphe et f(Δ) = Θ(Δ log2(Δ)). Leur résultat s'applique également à des modèles de spins généraux où la dépendance en Δ est nécessaire. Dans ce travail nous nous concentrons sur le modèle d'Ising ferromagnétique et montrons que le temps de mélange de la dynamique de Glauber est au moins de (1/4 + o(1))n log(n) sur n'importe quel graphe à n sommets.

Consider Glauber dynamics for the Ising model on a graph of n vertices. Hayes and Sinclair showed that the mixing time for this dynamics is at least nlog n/f(Δ), where Δ is the maximum degree and f(Δ) = Θ(Δlog2Δ). Their result applies to more general spin systems, and in that generality, they showed that some dependence on Δ is necessary. In this paper, we focus on the ferromagnetic Ising model and prove that the mixing time of Glauber dynamics on any n-vertex graph is at least (1/4 + o(1))nlog n.

DOI : https://doi.org/10.1214/10-AIHP402
Classification : 60J10,  60K35,  68W20
Mots clés : Glauber dynamics, mixing time, Ising model
     author = {Ding, Jian and Peres, Yuval},
     title = {Mixing time for the Ising model : a uniform lower bound for all graphs},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1020--1028},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {4},
     year = {2011},
     doi = {10.1214/10-AIHP402},
     zbl = {1274.82012},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/10-AIHP402/}
Ding, Jian; Peres, Yuval. Mixing time for the Ising model : a uniform lower bound for all graphs. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1020-1028. doi : 10.1214/10-AIHP402. http://www.numdam.org/articles/10.1214/10-AIHP402/

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