Diffusive scaling of the Kob-Andersen model in ${\mathbb{Z}}^d$

Abstract

We consider the Kob-Andersen model, a cooperative lattice gas with kinetic constraints which has been widely analyzed 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 $T_{\mathrm{rel}}$ of the system. Furthermore, as $q\downarrow 0$ we prove upper and lower bounds on $L^{-2}{T}_{\mathrm{rel}}(q,L)$ 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.

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