  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.
