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

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 stationary distribution of the KCIP Markov chain is the Binomial($\mathrm{\mid }G\mathrm{\mid },p$) distribution on the number of particles, 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 $G={\mathbb{Z}}_L^d$, $d \geq 3$,inthelow-densityregime$p = \frac{c}{n}$forarbitrary$0 < c < 1$; 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 modifcation of the conjecture, and show that this modifcation is correct up to logarithmic factors. The methods developed in this paper also provide a strategy for tackling Aldous’ conjecture for other graphs.