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

TL;DR

This paper proves a cutoff phenomenon for the unbiased one-sided transposition shuffle on generalized symmetric groups, extending previous results and laying groundwork for future biased shuffle analyses.

Contribution

It establishes the cutoff in total variation and separation distances for any fixed m ≥ 1, generalizing prior work and providing foundational branching rules.

Findings

Proves cutoff for unbiased OST shuffle on G_{m,n} in n log(n) time.

Establishes branching rules for simple modules of G_{m,n}.

Lays groundwork for cutoff conjecture in biased OST shuffles.

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 $G_{m,n}$.