Mixing times of one-sided $k$-transposition shuffles

TL;DR

This paper investigates the mixing times of the one-sided $k$-transposition shuffle, revealing slow mixing behavior even for large $k$, and explores cutoff phenomena using advanced eigenvector techniques.

Contribution

It introduces new analysis of the mixing times for the one-sided $k$-transposition shuffle and applies the lifting eigenvectors method to study cutoff behavior.

Findings

The shuffle mixes slowly even for large $k$

Different mixing behaviors are identified depending on $k$

Cutoff phenomena are explored using eigenvector techniques

Abstract

We study mixing times of the one-sided $k$-transposition shuffle. We prove that this shuffle mixes relatively slowly, even for $k$ big. Using the recent "lifting eigenvectors" technique of Dieker and Saliola and applying the $\ell^2$ bound, we prove different mixing behaviors and explore the occurrence of cutoff depending on $k$.