Mixing of the upper triangular matrix walk

TL;DR

This paper analyzes the mixing time of a random walk on upper triangular matrices over ield ield, showing it is or lazy walk is or matrices over ield ield, with proofs leveraging the group's linear structure.

Contribution

It establishes the optimal or the mixing time of a natural random walk on upper triangular matrices over ield ield, extending results to prime fields.

Findings

Mixing time is or the lazy walk on upper triangular matrices.

The proof leverages the linear structure of the matrix group.

Results extend to matrices over ield ield for prime ield ield.

Abstract

We study a natural random walk over the upper triangular matrices, with entries in the field $\Z_2$, generated by steps which add row $i+1$ to row $i$. We show that the mixing time of the lazy random walk is $O(n^2)$ which is optimal up to constants. Our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields $\Z_q$ for $q$ prime.