Cutoff of the simple exclusion process with inhomogeneous conductances

In this paper, we study the mixing time of the simple exclusion process with $k$ particles in the line segment $[1,N]$ with conductances $c^{(N)}(x,x+1{)}_{1\le x<N}$ where $c^{(N)}(x,x+1)>0$ is the rate of swapping the contents of the two sites $x$ and $x+1$. Writing $r^{(N)}(x,x+1):=1\mathrm{/}{c}^{(N)}(x,x+1)$, under the assumption \begin{equation*} \limsup_{N\to \infty}, \frac{1}{N}\sup_{1< m \le N}, \left| \sum_{x=2}^m r^{(N)}(x-1, x)- (m-1) \right|;=;0,, \end{equation*} and some further assumptions on $r^{(N)}(x,x+1{)}_{x\in \mathbb{N}}$ and $k$, we prove that around time $(1+o(1))(2{\pi }^2{)}^{-1}{N}^2\log k$, the total variation distance to equilibrium of the simple exclusion process drops abruptly from $1$ to $0$.