We obtain a tight bound of O(L2 log k) for the mixing time of the exclusion process in Zd/LZd with k < = 12 Ld particles. Previously the best bound, based on the log Sobolev constantdetermined by Yau, was not tight for small k. When dependence on the dimension d is consid-ered, our bounds are an improvement for all k. We also get bounds for the relaxation time thatare lower-order in d than previous estimates: our bound of O(L2 log d) improves on the earlierbound O(L2 d), obtained by Quastel. Our proof is based on an auxiliary Markov chain we callthe chameleon process, which may be of independent interest. Introduction Let G = (V, E) be a finite, connected graph and define a configuration as follows. In a configuration, each vertex in V contains either a black ball or a white ball (where balls of the same color are indistinguishable), and the number of black balls is at most |V |/2. The exclusion process on G is the following continuous-time Markov process on configurations. For each edge e at rate 1: switch the balls at the endpoints of e. Note that since the exclusion process is irreducible and has symmetric transition rates, the uniform distribution is stationary. Let C denote the space of configurations, and for probability distributions u, * on C, le
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