An Exercise (?) in Fourier Analysis on the Heisenberg Group

TL;DR

This paper investigates the convergence rate of a simple random walk on the Heisenberg group over Z/nZ, employing Fourier analysis to establish bounds and introduce novel spectral techniques applicable to various groups.

Contribution

It provides new spectral bounding techniques for random walks on groups, enhancing understanding of convergence times using Fourier analysis.

Findings

Random walk converges in order n^2 steps

Introduces novel spectral bounding techniques

Fourier analysis effectively analyzes group walks

Abstract

Let H(n) be the group of 3x3 uni-uppertriangular matrices with entries in Z/nZ, the integers mod n. We show that the simple random walk converges to the uniform distribution in order n^2 steps. The argument uses Fourier analysis and is surprisingly challenging. It introduces novel techniques for bounding the spectrum which are useful for a variety of walks on a variety of groups.