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 >= 4N(2)/9 pi(2) log N, we show that the probability density P-t (chi, center dot) is given by an exponential tilt of the equilibrium measure by the main eigenfunction of the particle system. As 4N(2)/9 pi(2) log N is smaller than the mixing time which is 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 -> infinity) d(N) (N-2/2 pi(2) log N + N-2/pi(2)s) = erf (root 2/pi e(-s)) where erf is the Gauss error function.