Cutoff for the asymmetric riffle shuffle

Abstract

In the Gilbert–Shannon–Reeds shuffle, a deck of N cards is cut into two approximately equal parts which are riffled together uniformly at random. Bayer and Diaconis (Ann. Appl. Probab. 2 294–313) famously showed that this Markov chain undergoes cutoff in total variation after $\frac{3log(\mathit{N})}{2log(2)}$ shuffles. We establish cutoff for the more general asymmetric riffle shuffles in which one cuts the deck into differently sized parts. The value of the cutoff point confirms a conjecture of Lalley from 2000 (Ann. Appl. Probab. 10 1302–1321). Some appealing consequences are that asymmetry always slows mixing and that total variation mixing is strictly faster than separation and ${\mathit{L}}^{\infty }$ mixing.