Spectral distributions of periodic random matrix ensembles

Kolo\u{g}lu, Kopp and Miller compute the limiting spectral distribution of a certain class of random matrix ensembles, known as $m$-block circulant ensembles, and discover that it is exactly equal to the eigenvalue distribution of an $m \times m$ Gaussian unitary ensemble. We give a simpler proof that under very general conditions which subsume the cases studied by Kolo\u{g}lu-Kopp-Miller, block-periodic symmetric ensembles always have limiting spectral distribution equal to the eigenvalue distribution of a finite ensemble which is a `complex version' of one of the blocks. The proofs show that this general correspondence between block-periodic random matrix ensembles and finite complex Hermitian ensembles is elementary and combinatorial in nature.