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 $(i, n and#8722; 1, n)$ and $(i, n, n and#8722; 1)$. 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 $(n and#8722; 3/2) \log n$ 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 $(and#8722;i, n)$ for $1 and#8804; i and#8804; n$. 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 $n \log n$... newline