Cutoff for the Asymmetric Riffle Shuffle

TL;DR

This paper proves the cutoff phenomenon for asymmetric riffle shuffles, extending previous results for symmetric shuffles and confirming Lalley's conjecture, with implications on mixing times and the effects of asymmetry.

Contribution

It establishes cutoff for asymmetric riffle shuffles, confirming Lalley's 2000 conjecture and analyzing how asymmetry affects mixing times.

Findings

Asymmetry always slows mixing.

Total variation mixing is faster than separation and $L^{}$ mixing.

Cutoff occurs at a specific time confirming theoretical predictions.

Abstract

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