Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Mixing times for a constrained Ising process on the two-dimensional torus at low density

Natesh S. Pillai and Aaron Smith

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Abstract

We study a kinetically constrained Ising process (KCIP) associated with a graph $G$ and density parameter $p$; this process is an interacting particle system with state space $\{0,1\}^{G}$, the locations of the particles. The ‘constraint’ in the name of the process refers to the rule that a vertex cannot change its state unless it has at least one neighbour in state ‘1’. The KCIP has been proposed by statistical physicists as a model for the glass transition. In this note, we study the mixing time of a KCIP on the 2-dimensional torus $G=\mathbb{Z}_{L}^{2}$ in the low-density regime $p=\frac{c}{L^{2}}$ for arbitrary $0<c<\infty$, extending our previous results for the analogous process on the torus $\mathbb{Z}_{L}^{d}$ in dimension $d\geq3$. Our general approach is similar, but the extension requires more delicate bounds on the behaviour of the process at intermediate densities.

Résumé

Nous étudions un processus d’Ising avec contraintes cinétiques (PICC) associé à un graphe $G$ et un paramètre de densité $p$. Ce processus est un système de particules en interaction avec espace d’états $\Omega=\{0,1\}^{G}$, décrivant les positions des particules. Les « contraintes » apparaissant dans le nom de ce processus réfèrent à la règle suivante: un sommet ne peut pas changer son état à moins qu’il ait un voisin dans l’état « 1 ». Le PICC a été proposé par des physiciens comme un modèle pour la transition vitreuse. Dans ce travail, nous analysons le temps de mélange d’un PICC sur le tore de dimension 2 $G=\mathbb{Z}_{L}^{2}$ dans le régime de faible densité $p=\frac{c}{L^{2}}$, où $0<c<\infty$. Ceci prolonge nos résultats au processus analogue sur le tore $G=\mathbb{Z}_{L}^{d}$, $d\geq3$. Notre approche générale est similaire, mais cette extension requiert des bornes plus délicates sur le comportement du processus aux densités intermédiaires.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1649-1678.

Dates
Received: 29 August 2017
Revised: 24 May 2018
Accepted: 4 September 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1569398881

Digital Object Identifier
doi:10.1214/18-AIHP930

Mathematical Reviews number (MathSciNet)
MR4010947

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]

Keywords
Kinetically constrained process Interacting particle systems Markov chain Mixing time Glass transition

Citation

Pillai, Natesh S.; Smith, Aaron. Mixing times for a constrained Ising process on the two-dimensional torus at low density. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1649--1678. doi:10.1214/18-AIHP930. https://projecteuclid.org/euclid.aihp/1569398881


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