The Annals of Probability

Mixing times for a constrained Ising process on the 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 location of the particles. The number of particles at stationarity follows the $\operatorname{Binomial}(|G|,p$) distribution, conditioned on having at least one particle. 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, and more recently as a simple algorithm for data storage in computer networks. In this note, we study the mixing time of this process on the torus $G=\mathbb{Z}_{L}^{d}$, $d\geq3$, in the low-density regime $p=\frac{c}{|G|}$ for arbitrary $0<c<\infty$; this regime is the subject of a conjecture of Aldous and is natural in the context of computer networks. Our results provide a counterexample to Aldous’ conjecture, suggest a natural modification of the conjecture, and show that this modification is correct up to logarithmic factors. The methods developed in this paper also provide a strategy for tackling Aldous’ conjecture for other graphs.

Article information

Source
Ann. Probab., Volume 45, Number 2 (2017), 1003-1070.

Dates
Received: January 2015
Revised: November 2015
First available in Project Euclid: 31 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1490947313

Digital Object Identifier
doi:10.1214/15-AOP1080

Mathematical Reviews number (MathSciNet)
MR3630292

Zentralblatt MATH identifier
1381.60105

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
Ising models interacting particle systems mixing times coalescent process north-east model

Citation

Pillai, Natesh S.; Smith, Aaron. Mixing times for a constrained Ising process on the torus at low density. Ann. Probab. 45 (2017), no. 2, 1003--1070. doi:10.1214/15-AOP1080. https://projecteuclid.org/euclid.aop/1490947313


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