Sums of twisted circulants

TL;DR

This paper analyzes the convergence rates of random walks on Heisenberg groups over finite rings, extending previous results to arbitrary generating sets and higher dimensions using Fourier analysis and eigenvalue bounds.

Contribution

It generalizes convergence rate results to arbitrary symmetric generators and higher-dimensional Heisenberg groups, employing Fourier analysis and eigenvalue bounds of twisted circulant matrices.

Findings

Convergence rates are established for various generating sets.

Eigenvalue bounds for sums of twisted circulant matrices are derived.

Results extend to higher-dimensional Heisenberg groups.

Abstract

The rate of convergence of simple random walk on the Heisenberg group over $Z/nZ$ with a standard generating set was determined by Bump et al [1,2]. We extend this result to random walks on the same groups with an arbitrary minimal symmetric generating set. We also determine the rate of convergence of simple random walk on higher-dimensional versions of the Heisenberg group with a standard generating set. We obtain our results via Fourier analysis, using an eigenvalue bound for sums of twisted circulant matrices. The key tool is a generalization of a version of the Heisenberg Uncertainty Principle due to Donoho-Stark [4].