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

## TL;DR

This paper proves that the relaxation time of the Kob-Andersen model exhibits diffusive scaling in any dimension, with bounds that support the idea that large vacancy droplets drive the system's equilibration.

## Contribution

It establishes the diffusive scaling of relaxation time for the Kob-Andersen model in any dimension, improving previous results and confirming physicists' hypotheses about the role of large vacancy droplets.

## Key findings

- Relaxation time scales diffusively with system size.
- Bounds on relaxation time as vacancy density approaches zero.
- Supports the theory of cooperative droplet motion as the main relaxation mechanism.

## 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 $T_{\rm rel}$ of the system. Furthermore, as $q\downarrow 0$ we prove upper and lower bounds on $L^{-2} T_{\rm rel} (q,L)$ 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.