Cutoff for product replacement on finite groups

Abstract

We analyze a Markov chain, known as the product replacement chain, on the set of generating $n$-tuples of a fixed finite group $G$. We show that as $n\to \infty$, the total-variation mixing time of the chain has a cutoff at time $\frac{3}{2}n\log n$ with window of order $n$. This generalizes a result of Ben-Hamou and Peres (who established the result for $G=\mathbb{Z}/2$) and confirms a conjecture of Diaconis and Saloff-Coste that for an arbitrary but fixed finite group, the mixing time of the product replacement chain is $O(n\log n)$.

26 pages, 1 figure