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Cutoff in total variation for the shelf shuffle

Andrea Ottolini, Ray Chen

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.

Cutoff in total variation for the shelf shuffle

Abstract

We analyze the mixing time of a popular shuffling machine known as the shelf shuffler. It is a modified version of a -handed riffle shuffle ( in casinos) in which a deck of cards is split multinomially into piles, the even-numbered piles are reversed, and then cards are dropped from piles proportionally to their sizes. We prove that 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.

Paper Structure

This paper contains 8 sections, 3 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Related literature
  4. Structure of the paper
  5. Number of valleys
  6. Monotonicity results
  7. On the mean and variance
  8. Proof of main theorem

Key Result

Theorem 1.1

Fix $c>0$. Then, where $\Phi(x) = \int_{-\infty}^{x} e^{-t^2 / 2}\:dt / \sqrt{2 \pi}$ is the cumulative distribution function of a standard normal random variable.

Figures (1)

  • Figure 1: Total variation after an $m$-shuffle on a $52$-card deck for $1 \le m \le 300$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}