# Total variation cutoff for the flip-transpose top with random shuffle

## TL;DR

This paper analyzes a specific random walk on the hyperoctahedral group, determining its spectral properties, mixing time, and cutoff phenomenon, and extends the results to a related group, providing insights into their convergence behaviors.

## Contribution

It explicitly finds the spectrum of the transition matrix and proves the cutoff phenomenon for the flip-transpose top shuffle on $B_n$, a novel analysis in this context.

## Key findings

- Mixing time is of order $n \,\log n$.
- The shuffle exhibits the cutoff phenomenon.
- Similar results are shown for the demihyperoctahedral group $D_n$.

## Abstract

We consider a random walk on the hyperoctahedral group $B_n$ generated by the signed permutations of the forms $(i,n)$ and $(-i,n)$ for $1\leq i\leq n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order $n\log n$. We also show that this shuffle exhibits the cutoff phenomenon. In the appendix, we show that a similar random walk on the demihyperoctahedral group $D_n$ also has a cutoff at $\left(n-\frac{1}{2}\right)\log n$.