Total variation cutoff for the transpose top-$2$ with random shuffle

TL;DR

This paper analyzes a specific random walk on the alternating group generated by certain 3-cycles, determining its spectral properties, mixing time, and establishing a total variation cutoff.

Contribution

It provides the spectrum of the transition matrix and proves a total variation cutoff for the transpose top-2 with random shuffle on the alternating group.

Findings

Spectral analysis of the transition matrix

Mixing time of order (n-1.5) log n

Existence of a total variation cutoff

Abstract

In this paper, we investigate the properties of a random walk on the alternating group $A_n$ generated by $3$-cycles of the form $(i,n-1,n)$ and $(i,n,n-1)$. We call this the transpose top-$2$ with random shuffle. We find the spectrum of the transition matrix of this shuffle. We show that the mixing time is of order $\left(n-\frac{3}{2}\right)\log n$ and prove that there is a total variation cutoff for this shuffle.