Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion
Résumé
In this paper, we investigate the mixing time of the adjacent transposition shuffle for a deck of cards. We prove that around time N^2\log N/(2\pi^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 (N^2\log N)/\pi^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.