Conjugacy-invariant random walks on nilpotent groups
Xiangying Huang
TL;DR
This paper studies mixing times of conjugacy-invariant random walks on finite nilpotent groups $G$ and shows that, in many natural regimes, mixing is dictated by the abelianization projection $G_{\mathrm{ab}}$. By decomposing the walk via the lower central series and exploiting a quotient structure, the authors prove a general bound: $t^{\mathrm{TV}}_{\mathrm{mix}}(G_{\mathrm{ab}},\varepsilon)\le t^{\mathrm{TV}}_{\mathrm{mix}}(G,\varepsilon)\le \max\{ t^{\ell^2}_{\mathrm{mix}}(G_{\mathrm{ab}},\varepsilon/2), \mu_*^{-1}(\log k+2\log(4/\varepsilon)) \}$, with $\mu_* = \min_a \mu(\mathrm{Cl}(s_a))$ and $k$ the number of conjugacy-class representatives in the support of $\mu$. This abelianization-dominated picture yields cutoff results for two conjugacy-invariant walks on $\mathbb{U}_n(p)$ with explicit cutoff times that match the abelian-projected rates, connecting to earlier work by Arias-Castro–Diaconis–Stanley and Nestoridi. The work thus reduces non-abelian mixing on nilpotent groups to an Abelian problem, offering a robust framework for analyzing a broad class of random walks with conjugacy-invariant jumps.
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).