The cutoff profile for the simple exclusion process on the circle

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

\[\lim_{N\to\infty}d_{N}(\frac{N^{2}}{2\pi^{2}}\log N+\frac{N^{2}}{\pi^{2}}s)=\operatorname{erf}(\frac{\sqrt{2}}{\pi}e^{-s}),\] where $\operatorname{erf}$ is the Gauss error function.