Cutoff in total variation for the shelf shuffle

TL;DR

This paper studies the mixing time of the shelf shuffler, a card shuffling method, proving precise bounds and the occurrence of a cutoff phenomenon with a detailed cutoff profile.

Contribution

It establishes the exact number of shuffles needed for mixing and characterizes the cutoff profile, advancing understanding of this specific shuffling process.

Findings

Necessary and sufficient shuffles: (5/4) log_{2m} n

Cutoff phenomenon with constant window size

Cutoff profile described by shifted normal distribution

Abstract

We analyze the mixing time of a popular shuffling machine known as the shelf shuffler. It is a modified version of a $2m$-handed riffle shuffle ($m=10$ in casinos) in which a deck of $n$ cards is split multinomially into $2m$ piles, the even-numbered piles are reversed, and then cards are dropped from piles proportionally to their sizes. We prove that $\frac{5}{4} \log_{2m} n$ shuffles are necessary and sufficient to mix in total variation, and a cutoff occurs with constant window size. We also determine the cutoff profile in terms of the total variation distance between two shifted normal random variables.