The mixing time for simple exclusion

We obtain a tight bound of $O(L^2\log k)$ for the mixing time of the exclusion process in ${\mathbf{Z}}^d\mathrm{/}L{\mathbf{Z}}^d$ with $k\le 1\mathrm{/}2L^d$ particles. Previously the best bound, based on the log Sobolev constant determined by Yau, was not tight for small $k$. When dependence on the dimension $d$ is considered, our bounds are an improvement for all $k$. We also get bounds for the relaxation time that are lower order in $d$ than previous estimates: our bound of $O(L^2\log d)$ improves on the earlier bound $O(L^{2}d)$ obtained by Quastel. Our proof is based on an auxiliary Markov chain we call the chameleon process, which may be of independent interest.