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March 2017 Mixing times for a constrained Ising process on the torus at low density
Natesh S. Pillai, Aaron Smith
Ann. Probab. 45(2): 1003-1070 (March 2017). DOI: 10.1214/15-AOP1080


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.


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Natesh S. Pillai. Aaron Smith. "Mixing times for a constrained Ising process on the torus at low density." Ann. Probab. 45 (2) 1003 - 1070, March 2017.


Received: 1 January 2015; Revised: 1 November 2015; Published: March 2017
First available in Project Euclid: 31 March 2017

zbMATH: 1381.60105
MathSciNet: MR3630292
Digital Object Identifier: 10.1214/15-AOP1080

Primary: 60J10
Secondary: 60J20

Keywords: Coalescent process , interacting particle systems , Ising models , Mixing times , north-east model

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 2 • March 2017
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