Spectral distributions of periodic random matrix ensembles

Kolo\u{g}lu, Kopp and Miller compute the limiting spectral distribution of a certain class of real random matrix ensembles, known as $k$-block circulant ensembles, and discover that it is exactly equal to the eigenvalue distribution of an $k \times k$ 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, real-symmetric ensembles with periodic diagonals always have limiting spectral distribution equal to the eigenvalue distribution of a finite Hermitian ensemble with Gaussian entries which is a `complex version' of a $k \times k$ submatrix of the ensemble. We also prove an essentially algebraic relation between certain periodic finite Hermitian ensembles with Gaussian entries, and the previous result may be seen as an asymptotic version of this for real-symmetric ensembles. The proofs show that this general correspondence between periodic random matrix ensembles and finite complex Hermitian ensembles is elementary and combinatorial in nature.