Conjugacy-invariant random walks on nilpotent groups

TL;DR

This paper provides bounds on the mixing times of conjugacy-invariant random walks on finite nilpotent groups, linking their behavior to the simpler abelianized projections, and demonstrates cutoff phenomena in specific matrix group examples.

Contribution

It introduces a comparison framework that relates nilpotent group walk mixing times to their abelianization projections, simplifying analysis.

Findings

Bounds on mixing times in terms of abelianization projections

Reduction of mixing analysis to Abelian setting

Establishment of cutoff phenomena in specific matrix groups

Abstract

We establish bounds on the mixing times of conjugacy-invariant random walks on finite nilpotent groups in terms of the mixing times of their projections onto the abelianization. This comparison framework shows that, in several natural cases of interest, the mixing behavior on a nilpotent group is governed by that of the projected walk on the abelianization, reducing the study of mixing to a simpler problem in the Abelian setting. As an application, these bounds yield cutoff for two examples of conjugacy-invariant walks on unit upper-triangular matrix groups previously studied by Arias-Castro, Diaconis, and Stanley (2004) and by Nestoridi (2019).