Upper triangular matrix walk: Cutoff for finitely many columns

TL;DR

This paper establishes a cutoff phenomenon for the mixing time of finitely many columns in the upper triangular matrix walk over _2, aligning it with the East process, and shows spectral gap equality.

Contribution

It proves a cutoff for the mixing of finitely many columns in the matrix walk, extending prior bounds and linking spectral gaps to the East process.

Findings

Cutoff occurs at the same time as the East process.

Spectral gaps of the matrix walk and East process are equal.

Recursive proof uses local dual processes and concentration results.

Abstract

We consider random walk on the group of uni-upper triangular matrices with entries in $\mathbb{F}_2$ which forms an important example of a nilpotent group. Peres and Sly (2013) proved tight bounds on the mixing time of this walk up to constants. It is well known that the single column projection of this chain is the one dimensional East process. In this article, we complement the Peres-Sly result by proving a cutoff result for the mixing of finitely many columns in the upper triangular matrix walk at the same location as the East process of the same dimension. Moreover, we also show that the spectral gaps of the matrix walk and the East process are equal. The proof of the cutoff result is based on a recursive argument which uses a local version of a dual process appearing in Peres and Sly (2013), various combinatorial consequences of mixing and concentration results for the movement of the front in the one dimensional East process.