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

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 $\underset{N\to \infty }{\lim }{\mu }_{X_N}$ 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 $\{\underset{N\to \infty }{\lim }{\mu }_{X_N}|m\in \mathbb{N}\}$ 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.

6 pages

References in corpus (1)

• The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices