Cutoff for the cyclic adjacent transposition shuffle

Abstract

We study the cyclic adjacent transposition (CAT) shuffle of $n$ cards, which is a systematic scan version of the random adjacent transposition (AT) card shuffle. In this paper, we prove that the CAT shuffle exhibits cutoff at $\frac{n^{3}}{2\pi^{2}}\log n$, which concludes that it is twice as fast as the AT shuffle. This is the first verification of cutoff phenomenon for a time-inhomogeneous card shuffle.