Symmetric exclusion versus independent random walks via the Octopus inequality

We prove that the total-variation $\varepsilon$-mixing time of the $k$-particle symmetric exclusion process is at most $34$ times the single-particle mixing time at error $\varepsilon/k$. The bound holds for every $\varepsilon\in(0,1)$ on every finite connected graph with symmetric edge rates, and $k$ may be replaced by the number of vacancies. This settles a conjecture of Oliveira. We also obtain bounds for the $L^2$ and $L^\infty$ mixing times in terms of independent walks, and a Dirichlet-form inequality with the complete graph whose coefficient is optimal up to a universal constant. Further consequences include logarithmic Sobolev inequalities and energy bounds for the spin-$1/2$ Heisenberg ferromagnet.