Diffusive scaling of the Kob–Andersen model inZ^d

We consider the Kob–Andersen model, a cooperative lattice gas with kinetic constraints which has been widely analysed in the physics literature in connection with the study of the liquid/glass transition. We consider the model in a finite box of linear size L with sources at the boundary. Our result, which holds in any dimension and significantly improves upon previous ones, establishes for any positive vacancy density q a purely diffusive scaling of the relaxation time Trel (q , L) of the system. Furthermore, as q ↓ 0 we prove upper and lower bounds on L−2Trel(q,L) which agree with the physicists belief that the dominant equilibration mechanism is a cooperative motion of rare large droplets of vacancies. The main tools combine a recent set of ideas and techniques developed to establish universality results for kinetically constrained spin models, with methods from bootstrap percolation, oriented percolation and canonical flows for Markov chains.