A note on eigenvalues of random block Toeplitz matrices with slowly growing bandwidth

TL;DR

This paper derives explicit density functions for the eigenvalue distribution of large random block Toeplitz matrices with slowly growing bandwidth, revealing a transition from normal to semicircular distributions and connecting to Gaussian ensembles.

Contribution

It provides explicit formulas for eigenvalue densities of large block Toeplitz matrices with slowly increasing bandwidth, linking them to Gaussian unitary and orthogonal ensembles.

Findings

Eigenvalue densities are explicitly derived for large matrices.

Discovered a transition from normal to semicircular eigenvalue distributions.

Established a relationship between block Toeplitz matrices and Gaussian ensembles.

Abstract

This paper can be thought of as a remark of \cite{llw}, where the authors studied the eigenvalue distribution $\mu_{X_N}$ of random block Toeplitz band matrices with given block order $m$. In this note we will give explicit density functions of $\lim\limits_{N\to\infty}\mu_{X_N}$ when the bandwidth grows slowly. In fact, these densities are exactly the normalized one-point correlation functions of $m\times m$ Gaussian unitary ensemble (GUE for short). The series $\{\lim\limits_{N\to\infty}\mu_{X_N}|m\in\mathbb{N}\}$ can be seen as a transition from the standard normal distribution to semicircle distribution. We also show a similar relationship between GOE and block Toeplitz band matrices with symmetric blocks.