Mixing Times for the Commuting Chain on CA Groups

Abstract

Let $G$ be a finite group. The commuting chain on $G$ moves from an element $x$ to $y$ by selecting $y$ uniformly amongst those which commute with $x$. The $t$ step transition probabilities of this chain converge to a distribution uniform on the conjugacy classes of $G$. We provide upper and lower bounds for the mixing time of this chain on a CA group (groups with a "nice" commuting structure) and show that cutoff does not occur for many of these chains. We also provide a formula for the characteristic polynomial of the transition matrix of this chain. We apply our general results to explicitly study the chain on several sequences of groups, such as general linear groups, Heisenberg groups, and dihedral groups. The commuting chain is a specific case of a more general family of chains known as Burnside processes. Few instances of the Burnside processes have permitted careful analysis of mixing. We present some of the first results on mixing for the Burnside process where the state space is not fully specified (i.e not for a particular group). Our upper bound shows our chain is rapidly mixing, a topic of interest for Burnside processes.