Total variation cutoff for random walks on some finite groups

Abstract

This thesis studies mixing times for three random walk models. Specifically, these are random walks on the alternating group, the group of signed permutations and the complete monomial group. The details for the models are given below:

The random walk on the alternating group: 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 obtain the sharp mixing time by proving the total variation cutoff phenomenon at $(n − 3/2) \log n$ for this shuffle.

The random walk on the group of signed permutations: We consider a random walk on the hyperoctahedral group $B_n$ generated by the signed permutations of the form $(i, n)$ and $(−i, n)$ for $1 ≤ i ≤ n$. 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 this shuffle exhibits the total variation cutoff phenomenon with cutoff time $n \log n$. Furthermore, we show that a similar random walk on the demihyperoctahedral group $D_n$ generated by the identity signed permutation and the signed permutations of the form $(i, n)$ and $(−i, n)$ for $1 ≤ i < n$ also has a cutoff at $(n − 1/2) \log n$.

The random walk on the complete monomial group: Let $G_1 ⊆ · · · ⊆ G_n ⊆ · · ·$ be a sequence of finite groups with $|G_1| > 2$. We study the properties of a random walk on the complete monomial group $G_n\wrS_n$ generated by the elements of the form $(e, . . . , e, g; id)$ and $(e, . . . , e, g^{−1} , e, . . . , e, g; (i, n))$ for $g ∈ G_n, 1 ≤ i < n$. 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 of order $n \log n + (1/2) n \log(|G_n | − 1)$. We also show that this shuffle satisfies cutoff phenomenon with cutoff time $n \log n$ if $|G_n| = o(n^{δ})$ for all $δ > 0$.