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

Abstract

We prove an upper bound for the $\varepsilon$-mixing time of the symmetric exclusion process on any graph $G$, with any feasible number of particles. Our estimate is proportional to ${\mathsf{T}}_{\mathsf{RW}(G)}\ln(|V|/\varepsilon )$, where $|V|$ is the number of vertices in $G$, and ${\mathsf{T}}_{\mathsf{RW}(G)}$ 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 Erdös–Rényi random graph and Poisson point processes in $\mathbb{R}^{d}$. Our technical tools include a variant of Morris’s chameleon process.