Cutoff for product replacement on finite groups

TL;DR

This paper proves that the product replacement Markov chain on finite groups exhibits a sharp cutoff at time  n  n , confirming a conjecture and generalizing previous results for specific groups.

Contribution

It establishes the cutoff phenomenon for the product replacement chain on arbitrary finite groups, extending prior work and confirming a conjecture about mixing times.

Findings

The chain has a cutoff at  n  n  time.

The mixing time is  n  n  with high probability.

The result applies to all fixed finite groups, not just specific cases.

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 $n \rightarrow \infty$, 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 $O(n \log n)$.