Super-character theory and comparison arguments for a random walk on the upper triangular matrices

TL;DR

This paper analyzes the mixing time of a specific random walk on upper triangular matrices over finite fields, showing it depends quadratically on the field size p, using super-character and comparison theories.

Contribution

It introduces a novel combination of super-character theory and comparison arguments to determine the mixing time dependence on p for the random walk.

Findings

Mixing time depends quadratically on p

Super-character theory effectively analyzes the walk

Comparison theory provides bounds on mixing time

Abstract

Consider the random walk on the $n \times n$ upper triangular matrices with ones on the diagonal and elements over $\mathbb{F}_p$ where we pick a row at random and either add it or subtract it from the row directly above it. The main result of this paper is to prove that the dependency of the mixing time on $p$ is $p^2$. This is proven by combining super-character theory and comparison theory arguments.