Cutoff for random walks on dihedral groups

TL;DR

This paper investigates the cutoff phenomenon for random walks on dihedral groups driven by uniform measures, revealing precise cutoff times and contrasting behavior with Abelian groups, while developing new analytical techniques.

Contribution

It establishes the occurrence of cutoff with high probability in a broad regime for dihedral groups and introduces sharper entropic estimation methods for related high-dimensional lattice processes.

Findings

Cutoff occurs with high probability for certain regimes of k.

The cutoff time differs from the entropic time known for Abelian groups.

New techniques for entropic estimates on high-dimensional dependent processes.

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.