Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cutoff for random walks on dihedral groups

Xiangying Huang, Renyu Rao

TL;DR

This work establishes a cutoff phenomenon for random walks on finite dihedral groups driven by $k$ i.i.d. uniform generators in the broad regime $1 \ll \log k \ll \log |G|$, identifying the mixing/cutoff time $t_0(k,G)$ and proving high-probability abrupt convergence to uniformity. The authors develop an entropic framework centered on an auxiliary process that separates reflections and rotations, and they derive sharp entropy concentration results for dependent coordinates via multivariate normal approximations and Chung–Diaconis–Graham-type analyses. In the regime $k \gg \log|G|$, the cutoff result extends to random Cayley graphs of virtually Abelian groups, with the same time scale $t_0(k,G)$, highlighting a universality beyond strictly Abelian settings. The technical core is sharp entropy control for the auxiliary process across regimes, enabling precise lower and upper bounds on mixing times and revealing the nuanced role of group structure (Abelianization) in non-Abelian contexts. These techniques advance the understanding of cutoff universality for virtually Abelian groups and may apply to broader high-dimensional dependent-coordinate models.

Abstract

We study the random walk on a finite dihedral group $G$ driven by the uniform measure on $k$ independently and uniformly chosen elements. We show that the walk exhibits cutoff with high probability throughout nearly the entire regime $1 \ll \log k \ll \log |G|$, and determine the precise cutoff time. Interestingly, this mixing time differs from the entropic time that characterizes cutoff behavior for random walks on Abelian groups. When $k \gg \log|G|$ and $\log k \ll \log|G|$, cutoff occurs with high probability on random Cayley graphs of virtually Abelian groups. The analysis develops techniques for obtaining sharper entropic estimates of an auxiliary process on high-dimensional lattices with dependent coordinates, which may also prove useful for related models in broader contexts.

Cutoff for random walks on dihedral groups

TL;DR

This work establishes a cutoff phenomenon for random walks on finite dihedral groups driven by i.i.d. uniform generators in the broad regime , identifying the mixing/cutoff time and proving high-probability abrupt convergence to uniformity. The authors develop an entropic framework centered on an auxiliary process that separates reflections and rotations, and they derive sharp entropy concentration results for dependent coordinates via multivariate normal approximations and Chung–Diaconis–Graham-type analyses. In the regime , the cutoff result extends to random Cayley graphs of virtually Abelian groups, with the same time scale , highlighting a universality beyond strictly Abelian settings. The technical core is sharp entropy control for the auxiliary process across regimes, enabling precise lower and upper bounds on mixing times and revealing the nuanced role of group structure (Abelianization) in non-Abelian contexts. These techniques advance the understanding of cutoff universality for virtually Abelian groups and may apply to broader high-dimensional dependent-coordinate models.

Abstract

We study the random walk on a finite dihedral group driven by the uniform measure on independently and uniformly chosen elements. We show that the walk exhibits cutoff with high probability throughout nearly the entire regime , and determine the precise cutoff time. Interestingly, this mixing time differs from the entropic time that characterizes cutoff behavior for random walks on Abelian groups. When and , cutoff occurs with high probability on random Cayley graphs of virtually Abelian groups. The analysis develops techniques for obtaining sharper entropic estimates of an auxiliary process on high-dimensional lattices with dependent coordinates, which may also prove useful for related models in broader contexts.
Paper Structure (25 sections, 32 theorems, 283 equations)

This paper contains 25 sections, 32 theorems, 283 equations.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. Statement of results
  4. Overview and Methodology
  5. Literature on the universality of cutoff
  6. Preliminaries
  7. Representation of the random walk
  8. Entropy of Auxiliary Process
  9. Entropy of $\mathbf{C}_\mathrm{R}$
  10. Entropy Asymptotics for $\mathbf{C}_\mathrm{S}$
  11. Definitions and notations
  12. Asymptotics of entropy
  13. Regime $k\gg 1$ with $k^2\ll |G|^{2/k}$
  14. Concentration of entropy for multivariate normal
  15. Entropy of auxiliary process
  16. ...and 10 more sections

Key Result

Theorem 1.3

Let $G = G^{(n)}$ be a sequence of finite dihedral groups as in dihedral_group with prime $n$, and let $k = k_n$ diverge with $n$. Let $\mathcal{S} = \{Z_i^{\pm 1} : i \in [k]\}$, where $Z_1, \dots, Z_k$ are i.i.d. uniform elements of $G$. Assume that $1 \ll \log k \ll \log |G|$. Within this range, then the random walk on $\mathrm{Cay}(G, \mathcal{S})$ exhibits cutoff with high probability at the

Theorems & Definitions (68)

  • Conjecture 1: Aldous and Diaconis, 1985 AldousDiaconis1985techreport
  • Definition 1.1: Undirected Cayley graph
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 1.5: Lemma 2.6 of hermon2021cutoff
  • Proposition 1.6: Proposition A.2 hermon2018supplementary
  • Proposition 2.2
  • proof
  • Definition 2.3: Filtrations
  • ...and 58 more