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Mixing time of the torus shuffle

Olena Blumberg, Ben Morris, Alto Senda

Abstract

We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only triplets of cards. Then we use it to analyze a classic model of card shuffling. In 1988, Diaconis introduced the following Markov chain. Cards are arranged in an $n$ by $n$ grid. Each step, choose a row or column, uniformly at random, and cyclically rotate it by one unit in a random direction. He conjectured that the mixing time is ${\rm O}(n^3 \log n)$. We obtain a bound that is within a poly log factor of the conjecture.

Mixing time of the torus shuffle

Abstract

We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only triplets of cards. Then we use it to analyze a classic model of card shuffling. In 1988, Diaconis introduced the following Markov chain. Cards are arranged in an by grid. Each step, choose a row or column, uniformly at random, and cyclically rotate it by one unit in a random direction. He conjectured that the mixing time is . We obtain a bound that is within a poly log factor of the conjecture.

Paper Structure

This paper contains 11 sections, 11 theorems, 155 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Conditional Relative Entropies
  4. Collisions and Monte shuffles
  5. $3$-collisions and entropy decomposition
  6. $3$-collisions/entropy theorem
  7. The Torus Shuffle
  8. Communicating classes
  9. The 3-stage procedure
  10. Applying Theorem \ref{['maintheorem']} to bound the mixing time
  11. Acknowledgments

Key Result

Theorem 4.1

Let $\pi$ be a 3-Monte shuffle on $n$ cards. Fix an integer $t > 0$ and suppose that $T$ is a random variable taking values in $\{1, \dots,t\}$ that is independent of the shuffles $\{\pi_i : i \geq 0\}$. For distinct cards $x, y$ and $z$, let $T_{xyz}$ be the time of the first $3$-collision in the t where $E_k = {\mathbf E}({\rm ENT}(\mu^{-1}(k) {\,|\,} {\mathcal{\tilde{T}}}_{k+1}^\mu))$ and $C$ i

Figures (7)

  • Figure 1: A possible move in the torus shuffle in the $4 \times 4$ case. Here, the second row is moved to the right
  • Figure 2: Labeling the grid in the case $n=4$.
  • Figure 3: A possible 3-collision. The first sequence shows the 2nd row shifted right, then the 2nd column shifted down. The second sequence shows the 2nd column shifted down, then the 2nd row shifted right. Note how only the shaded tiles differ in the end result by a 3-cycle.
  • Figure 4: The three-stage procedure.
  • Figure 5: The boxes involved in the 3-Stage procedure.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Theorem 4.1
  • proof
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 6.1
  • proof : Proof of Lemma \ref{['mainlemma']}
  • Theorem 6.2
  • Lemma 6.3
  • proof
  • Lemma 6.4
  • ...and 10 more