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 $N^2\log N/(2\pi^2)$, 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.