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.
Martinelli, F., Shapira, A.A., Toninelli, C. (2020). Diffusive scaling of the Kob–Andersen model inZ^d. ANNALES DE L'INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, 56(3), 2189-2210 [10.1214/19-AIHP1035].