Random walk on unipotent matrix groups

TL;DR

This paper introduces a new method for proving central limit theorems for random walks on nilpotent groups, with applications to the Heisenberg group and uni-upper triangular groups, providing sharp estimates for coordinate mixing times.

Contribution

The paper presents a novel approach for establishing CLTs on nilpotent groups, relaxing conditions and precisely analyzing coordinate mixing times.

Findings

Established a local CLT on the Heisenberg group with weaker assumptions.

Derived sharp bounds on the number of steps needed for randomness in matrix coordinates.

Provided explicit mixing time estimates for coordinates on the uni-upper triangular group.

Abstract

We introduce a new method for proving central limit theorems for random walk on nilpotent groups. The method is illustrated in a local central limit theorem on the Heisenberg group, weakening the necessary conditions on the driving measure. As a second illustration, the method is used to study walks on the $n\times n$ uni-upper triangular group with entries taken modulo $p$. The method allows sharp answers to the behavior of individual coordinates: coordinates immediately above the diagonal require order $p^2$ steps for randomness, coordinates on the second diagonal require order $p$ steps; coordinates on the $k$th diagonal require order $p^{\frac{2}{k}}$ steps.