Cutoff for the Adjacent Transposition Shuffle on a Cycle

In the adjacent transposition shuffle on a cycle, $n$ distinct cards are placed at the vertices of a cycle, and adjacent cards swap positions according to independent Poisson clocks of rate 1. We prove that this Markov chain exhibits total variation cutoff at time $n^2\log n\mathrm{/}(8{\pi }^2)$, with a window of order at most $n^2\log \log n$.