Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Mixing time for the Ising model: A uniform lower bound for all graphs

Jian Ding and Yuval Peres

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 4 (2011), 1020-1028.

Dates
First available in Project Euclid: 6 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1317906499

Digital Object Identifier
doi:10.1214/10-AIHP402

Mathematical Reviews number (MathSciNet)
MR2884222

Zentralblatt MATH identifier
1274.82012

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 68W20: Randomized algorithms

Keywords
Glauber dynamics Mixing time Ising model

Citation

Ding, Jian; Peres, Yuval. Mixing time for the Ising model: A uniform lower bound for all graphs. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 4, 1020--1028. doi:10.1214/10-AIHP402. https://projecteuclid.org/euclid.aihp/1317906499


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