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

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

2011-08-13

arxiv.org/abs/1108.2810v1

Mathematics

Probability

6 pages

Scientific paper

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.

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