Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion

In this paper, we investigate the mixing time of the adjacent transposition shuffle for a deck of N cards. We prove that around time N2logN/(2.2), the total variation distance to equilibrium of the deck distribution drops abruptly from 1 to 0, and that the separation distance has a similar behavior but with a transition occurring at time (N2logN)/.2. This solves a conjecture formulated by David Wilson. We present also similar results for the exclusion process on a segment of length N with k particles.