Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk

We prove an upper bound for the $\epsilon$-mixing time of the symmetric exclusion process on any graph G, with any feasible number of particles. Our estimate is proportional to ${\mathsf{T}}_{\mathsf{R}\mathsf{W}(G)}\ln (\mathrm{\mid }V\mathrm{\mid }\mathrm{/}\epsilon )$, where |V| is the number of vertices in G, and ${\mathsf{T}}_{\mathsf{R}\mathsf{W}(G)}$ is the 1/4-mixing time of the corresponding single-particle random walk. This bound implies new results for symmetric exclusion on expanders, percolation clusters, the giant component of the Erdos-Renyi random graph and Poisson point processes in ${\mathbb{R}}^d$. Our technical tools include a variant of Morris’s chameleon process.