Cutoff phenomenon for the warp-transpose top with random shuffle

TL;DR

This paper analyzes a specific random shuffle on the complete monomial group, determining its spectral properties and proving it exhibits cutoff phenomena in both  and total variation distances with explicit mixing times.

Contribution

It introduces the warp-transpose top with random shuffle on $G_n \u2283 S_n$, finds its spectrum, and establishes cutoff phenomena with precise mixing times.

Findings

Spectral analysis of the transition matrix.

Mixing time is $O(n \u2261  n + rac{1}{2} n \u2261  (|G_n|-1))$.

Exhibits -cutoff at $n \u2261  n + rac{1}{2} n \u2261  (|G_n|-1)$ and total variation cutoff at $n \u2261  n.

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 $n\log n$.