Mixing time for the asymmetric simple exclusion process in a random environment

Abstract

We consider the simple exclusion process in the integer segment $⟦1,\mathit{N}⟧$ with $\mathit{k}\le \mathit{N}/2$ particles and spatially inhomogenous jumping rates. A particle at site $\mathit{x}\in ⟦1,\mathit{N}⟧$ jumps to site $\mathit{x}-1$ (if $\mathit{x}\ge 2$) at rate $1-{\mathit{\omega }}_{\mathit{x}}$ and to site $\mathit{x}\mathbf{+}1$ (if $\mathit{x}\le \mathit{N}-1$) at rate ${\mathit{\omega }}_{\mathit{x}}$ if the target site is not occupied. The sequence $\mathit{\omega }={({\mathit{\omega }}_{\mathit{x}})}_{\mathit{x}\in \mathbb{Z}}$ is chosen by IID sampling from a probability law whose support is bounded away from zero and one (in other words the random environment satisfies the uniform ellipticity condition). We further assume $\mathbb{E}[log{\mathit{\rho }}_1]<0$ where ${\mathit{\rho }}_1:=(1-{\mathit{\omega }}_1)/{\mathit{\omega }}_1$, which implies that our particles have a tendency to move to the right. We prove that the mixing time of the exclusion process in this setup grows like a power of N. More precisely, for the exclusion process with ${\mathit{N}}^{\mathit{\beta }\mathbf{+}\mathit{o}(1)}$ particles where $\mathit{\beta }\in [0,1]$, we have in the large N asymptotic

$${\mathit{N}}^{max(1,\frac{1}{\mathit{\lambda }},\mathit{\beta }\mathbf{+}\frac{1}{2\mathit{\lambda }})\mathbf{+}\mathit{o}(1)}\le {\mathit{t}}_{\mathrm{mix}}^{\mathit{N},\mathit{k}}\le {\mathit{N}}^{\mathit{C}\mathbf{+}\mathit{o}(1)},$$

where $\mathit{\lambda }>0$ is such that $\mathbb{E}[{\mathit{\rho }}_1^{\mathit{\lambda }}]=1$ ($\mathit{\lambda }=\infty$ if the equation has no positive root) and C is a constant, which depends on the distribution of ω. We conjecture that our lower bound is sharp up to subpolynomial correction.