The cutoff profile for the simple exclusion process on the circle

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 $$\underset{N\to \mathrm{\infty }}{\lim }{d}_N(\frac{N^2}{2{\pi }^2}\log N+\frac{N^2}{{\pi }^2}s)=\mathrm{erf}\left(\frac{\sqrt{2}}{\pi }{e}^{-s}\right),$$ where $\mathrm{erf}$ is the Gauss error function.