Cutoff of the simple exclusion process with inhomogeneous conductances

TL;DR

This paper investigates the mixing time of the simple exclusion process with inhomogeneous conductances on a line segment, showing an abrupt transition from non-equilibrium to equilibrium around a specific time scale under certain conditions.

Contribution

It establishes a cutoff phenomenon for the mixing time of the exclusion process with inhomogeneous conductances, extending understanding of convergence behavior in non-uniform settings.

Findings

Mixing time exhibits a cutoff at approximately (2π²)^{-1} N² log k.

Abrupt transition from total variation distance 1 to 0.

Results depend on specific conditions on conductance ratios and particle number.

Abstract

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 $r^{(N)}(x, x+1) := 1/c^{(N)}(x, x+1)$, 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 $0$.