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 location of the particles. The number of particles at stationarity follows the 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=ZdL, d.3, in the low-density regime p=c|G| for arbitrary 0<c<.; 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.