This paper can be thought of as a remark of \cite{llw}, where the authors
studied the eigenvalue distribution ΞΌXNββ of random block Toeplitz band
matrices with given block order m. In this note we will give explicit density
functions of NββlimβΞΌXNββ when the bandwidth grows
slowly. In fact, these densities are exactly the normalized one-point
correlation functions of mΓm Gaussian unitary ensemble (GUE for short).
The series {NββlimβΞΌXNβββ£mβN} 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.Comment: 6 page