The exclusion process mixes (almost) faster than independent particles

TL;DR

This paper proves that the mixing time of the exclusion process on certain graphs is almost as fast as that of independent particles, providing new bounds and confirming parts of Oliveira's conjecture.

Contribution

It verifies Oliveira's conjecture up to a constant factor for specific classes of graphs and offers new mixing time bounds for various graph families.

Findings

Mixing time of exclusion process is nearly as fast as independent particles on certain graphs.

Provides probabilistic proof of a weaker version of Aldous' spectral-gap conjecture.

Establishes new mixing bounds for expanders, hypercubes, and vertex-transitive graphs.

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.