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

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.