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Random Walks on the Generalized Symmetric Group: Cutoff for the One-sided Transposition Shuffle

Yongtao Deng, Shi Jie Samuel Tan

Abstract

In this paper, we present a detailed proof for the exhibition of a cutoff for the one-sided transposition (OST) shuffle on the generalized symmetric group $G_{m,n}$. Our work shows that based on techniques for $m \leq 2$ proven by Matheau-Raven, we can prove the cutoff in total variation distance and separation distance for an unbiased OST shuffle on $G_{m,n}$ for any fixed $m \geq 1$ in time $n \log(n)$. We also prove the branching rules for the simple modules of $G_{m,n}$ and lay down some of the mathematical foundation for proving the conjecture for the cutoff in total variation distance for any general biased OST shuffle on $G_{m,n}$.

Random Walks on the Generalized Symmetric Group: Cutoff for the One-sided Transposition Shuffle

Abstract

In this paper, we present a detailed proof for the exhibition of a cutoff for the one-sided transposition (OST) shuffle on the generalized symmetric group . Our work shows that based on techniques for proven by Matheau-Raven, we can prove the cutoff in total variation distance and separation distance for an unbiased OST shuffle on for any fixed in time . We also prove the branching rules for the simple modules of and lay down some of the mathematical foundation for proving the conjecture for the cutoff in total variation distance for any general biased OST shuffle on .
Paper Structure (20 sections, 20 theorems, 74 equations)

This paper contains 20 sections, 20 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Random Walks on Finite Groups and Their Applications
  3. Main Result
  4. Discussion of Related and Further work
  5. Organization of the Paper
  6. Acknowledgements
  7. Preliminaries
  8. Random Walks on Finite Group
  9. Generalized Symmetric Groups
  10. One-sided Transposition Shuffle (OST)
  11. One-sided Transposition Shuffle on the Generalized Symmetric Group
  12. Lower Bound for OST Shuffle on The Generalized Symmetric Group.
  13. Upper Bound for OST Shuffle on The Generalized Symmetric Group.
  14. Cutoff for OST Shuffle on the Generalized Symmetric Group.
  15. Branching Rules for the Generalized Symmetric Group
  16. ...and 5 more sections

Key Result

Theorem 1.1

The unbiased one-sided transposition shuffle on the generalized symmetric group $G_{m,n}$ approaches the uniform distribution after $n\log(n)$ shuffles.

Theorems & Definitions (55)

  • Theorem 1.1: Conjectured in Section 4.5 in M20
  • Theorem 1.2: M20
  • Conjecture 1.3: M20
  • Conjecture 1.4: M20
  • Conjecture 1.5: M20
  • Conjecture 1.6: M20
  • Definition 2.1: MR0841111
  • Lemma 2.2: MR2466937
  • Definition 2.3: MR3530321
  • Theorem 2.4: MR2466937, the Convergence Theorem
  • ...and 45 more