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We study a kinetically constrained Ising process (KCIP) associated with agraph G and density parameter p; this process is an interacting particle systemwith state space $\{0,1\}^{G}$. The stationary distribution of the KCIP Markovchain is the Binomial($|G|, p$) distribution on the number of particles,conditioned on having at least one particle. The `constraint' in the name ofthe process refers to the rule that a vertex cannot change its state unless ithas at least one neighbour in state `1'. The KCIP has been proposed bystatistical physicists as a model for the glass transition, and more recentlyas a simple algorithm for data storage in computer networks. In this note, westudy the mixing time of this process on the torus $G = \mathbb{Z}_{L}^{d}$, $d\geq 3$, in the low-density regime $p = \frac{c}{n}$ for arbitrary $0 < c < 1$;this regime is the subject of a conjecture of Aldous and is natural in thecontext of computer networks. Our results provide a counterexample to Aldous'conjecture, suggest a natural modifcation of the conjecture, and show that thismodifcation is correct up to logarithmic factors. The methods developed in thispaper also provide a strategy for tackling Aldous' conjecture for other graphs.

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