Cutoff phenomenon for the warp-transpose top with random shuffle

Let $\{G_n\}_1^{\infty}$ be a sequence of non-trivial finite groups. In this paper, we study the properties of a random walk on the complete monomial group $G_n\wr S_n$ generated by the elements of the form $(\text{e},\dots,\text{e},g;\text{id})$ and $(\text{e},\dots,\text{e},g^{-1},\text{e},\dots,\text{e},g;(i,n))$ for $g\in G_n,\;1\leq i< n$. We call this the warp-transpose top with random shuffle on $G_n\wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is $O\left(n\log n+\frac{1}{2}n\log (|G_n|-1)\right)$. We show that this shuffle exhibits $\ell^2$-cutoff at $n\log n+\frac{1}{2}n\log (|G_n|-1)$ and total variation cutoff at $n\log n$.