Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cutoff phenomenon for the warp-transpose top with random shuffle

Subhajit Ghosh

TL;DR

The paper analyzes a warp-transpose top with random shuffle on the complete monomial group $G_n\wr S_n$, deriving its spectral data via non-commutative Fourier analysis on wreath products. It proves a mixing time of $O\left(n\log n+\tfrac{1}{2}n\log(|G_n|-1)\right)$, establishes an $\ell^2$-cutoff at that time, and shows a total-variation cutoff at $n\log n$, highlighting a regime where spectral bounds fail when $\log(|G_n|-1)$ is not $o(\log n)$. The results generalize prior Shuffle analyses (including transpose-top variants) to arbitrary finite groups $G_n$ and reveal how the group size influences the cutoff landscape. The work thus provides precise mixing profiles for a broad class of random walks on wreath products, with implications for spectral methods and coupling techniques in finite-group Markov chains.

Abstract

Let $\{G_n\}_1^{\infty}$ be a sequence of non-trivial finite groups. In this paper, we study the properties of a random walk on the complete monomial group $G_n\wr S_n$ generated by the elements of the form $(\text{e},\dots,\text{e},g;\text{id})$ and $(\text{e},\dots,\text{e},g^{-1},\text{e},\dots,\text{e},g;(i,n))$ for $g\in G_n,\;1\leq i< n$. We call this the warp-transpose top with random shuffle on $G_n\wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is $O\left(n\log n+\frac{1}{2}n\log (|G_n|-1)\right)$. We show that this shuffle exhibits $\ell^2$-cutoff at $n\log n+\frac{1}{2}n\log (|G_n|-1)$ and total variation cutoff at $n\log n$.

Cutoff phenomenon for the warp-transpose top with random shuffle

TL;DR

The paper analyzes a warp-transpose top with random shuffle on the complete monomial group , deriving its spectral data via non-commutative Fourier analysis on wreath products. It proves a mixing time of , establishes an -cutoff at that time, and shows a total-variation cutoff at , highlighting a regime where spectral bounds fail when is not . The results generalize prior Shuffle analyses (including transpose-top variants) to arbitrary finite groups and reveal how the group size influences the cutoff landscape. The work thus provides precise mixing profiles for a broad class of random walks on wreath products, with implications for spectral methods and coupling techniques in finite-group Markov chains.

Abstract

Let be a sequence of non-trivial finite groups. In this paper, we study the properties of a random walk on the complete monomial group generated by the elements of the form and for . We call this the warp-transpose top with random shuffle on . We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is . We show that this shuffle exhibits -cutoff at and total variation cutoff at .

Paper Structure

This paper contains 8 sections, 21 theorems, 111 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Representation theory of finite group
  3. Discrete time Markov chain with finite state space
  4. Non-commutative Fourier analysis and random walks on finite groups
  5. Spectrum of the transition matrix
  6. Order of the mixing time and $\ell^2$-cutoff
  7. Upper bound for total variation distance
  8. Lower bound for total variation distance

Key Result

Theorem 1.1

The $($total variation and $\ell^2\text{-})$ mixing time for the warp-transpose top with random shuffle on $\mathop{\mathrm{\mathcal{G}_n}}\nolimits$ is $O\left(n\log n+\frac{1}{2}n\log (|G_n|-1)\right)$.

Figures (6)

  • Figure 1: Transitions for the warp-transpose top with random shuffle on $\mathbb{Z}_3\wr S_{9}$. $\mathbb{Z}_3$ is the additive group of integers modulo $3$, consists of the colours red, green and blue such that red represents the identity element. $(a)$ shows transitions when the sixth card is chosen and $(b)$ shows transitions when the last card is chosen.
  • Figure 2: Young diagrams with $4$ boxes.
  • Figure 3: An Young $\mathbb{Z}_{10}$-diagram with $10$ boxes $\mu$. Here $\widehat{\mathbb{Z}}_{10}:=\{\sigma_i:1\leq i\leq 10\}$.
  • Figure 4: Standard Young tableaux of shape $(3,1)$.
  • Figure 5: A standard Young $\mathbb{Z}_{10}$-tableaux of shape $\mu$, defined in Figure \ref{['fig:Young_G_diagram']}.
  • ...and 1 more figures

Theorems & Definitions (50)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.4
  • proof
  • Definition 2.1
  • ...and 40 more