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 3log(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 L∞ mixing.