The exclusion process mixes (almost) faster than independent particles

Abstract

Oliveira conjectured that the order of the mixing time of the exclusion process with $k$-particles on an arbitrary $n$-vertex graph is at most that of the mixing-time of $k$ independent particles. We verify this up to a constant factor for $d$-regular graphs when each edge rings at rate $1/d$ in various cases:

(1) when $d=\Omega (\log _{n/k}n)$,

(2) when $\mathrm{gap}:=$ the spectral-gap of a single walk is $O(1/\log ^{4}n)$ and $k\ge n^{\Omega (1)}$,

(3) when $k\asymp n^{a}$ for some constant $0<a<1$.

In these cases, our analysis yields a probabilistic proof of a weaker version of Aldous’ famous spectral-gap conjecture (resolved by Caputo et al.). We also prove a general bound of $O(\log n\log \log n/\mathrm{gap})$, which is within a $\log \log n$ factor from Oliveira’s conjecture when $k\ge n^{\Omega (1)}$. As applications, we get new mixing bounds:

(a) $O(\log n\log \log n)$ for expanders,

(b) order $d\log (dk)$ for the hypercube $\{0,1\}^{d}$,

(c) order $(\mathrm{Diameter})^{2}\log k$ for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus.