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

This paper can be thought of as a remark of Li et al. (2010), where the authors studied the eigenvalue distribution μXNμXN of random block Toeplitz band matrices with given block order mm. In this paper, we will give explicit density functions of limN→∞μXNlimN→∞μXN when the bandwidth grows slowly. In fact, these densities are exactly the normalized one-point correlation functions of m×mm×m Gaussian unitary ensemble (GUE for short). The series {limN→∞μXN∣m∈N}{limN→∞μXN∣m∈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.