The cutoff profile for the simple exclusion process on the circle

TL;DR

This paper precisely characterizes the cutoff profile for the simple exclusion process on a circle, showing how the system relaxes to equilibrium and providing a sharp description of the total-variation distance decay.

Contribution

It offers an accurate description of the relaxation to equilibrium for the exclusion process, including the cutoff profile and the exponential tilt of the equilibrium measure.

Findings

The cutoff occurs at (N^2/2π^2) log N with a specific profile.

The total-variation distance converges to an error function profile.

The relaxation is described by an exponential tilt of the equilibrium measure.

Abstract

In this paper, we give a very accurate description of the way the simple exclusion process relaxes to equilibrium. Let $P_t$ denote the semi-group associated the exclusion on the circle with $2N$ sites and $N$ particles. For any initial condition $\chi$, and for any $t\ge\frac{4N^2}{9\pi ^2}\log N$, we show that the probability density $P_t(\chi,\cdot)$ is given by an exponential tilt of the equilibrium measure by the main eigenfunction of the particle system. As $\frac{4N^2}{9\pi^2}\log N$ is smaller than the mixing time which is $\frac{N^2}{2\pi^2}\log N$, this allows to give a sharp description of the cutoff profile: if $d_N(t)$ denote the total-variation distance starting from the worse initial condition we have \[\lim_{N\to\infty}d_N\biggl(\frac{N^2}{2\pi^2}\log N+\frac{N^2}{\pi^2}s\biggr)=\operatorname {erf}\biggl(\frac{\sqrt{2}}{\pi}e^{-s}\biggr),\] where $\operatorname {erf}$ is the Gauss error function.