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

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.

22 pages, 1 table, 1 figure. Minor revisions. The main result (Theorem 3.2 in the older version) has been stated (Theorem 1.1 in the updated version) in the introduction. The link between $P$ and the Jucys Murphy elements has been presented in Lemma 2.1. Examples of Corollary 2.6 have been given in section 2. The figure has been updated from $n=8$ to $n=10$. Some minor changes are done