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 Pt denote the semi-group associated the exclusion on the circle with 2N sites and N particles. For any initial condition ., and for any t.4N29.2logN, we show that the probability density Pt(.,.) is given by an exponential tilt of the equilibrium measure by the main eigenfunction of the particle system. As 4N29.2logN is smaller than the mixing time which is N22.2logN, this allows to give a sharp description of the cutoff profile: if dN(t) denote the total-variation distance starting from the worse initial condition we have limN..dN(N22.2logN+N2.2s)=erf(2...e.s), where erf is the Gauss error function.