Spectral distributions of periodic random matrix ensembles

TL;DR

This paper provides a simplified proof that periodic real symmetric random matrix ensembles have limiting spectral distributions matching those of finite complex Hermitian ensembles, revealing a fundamental algebraic and combinatorial connection.

Contribution

It introduces a general, elementary proof linking periodic real symmetric ensembles to finite complex Hermitian ensembles, extending prior specific results.

Findings

Limiting spectral distribution equals eigenvalue distribution of a finite Hermitian ensemble.

Established an algebraic relation between periodic Hermitian ensembles with Gaussian entries.

Proved the connection is elementary and combinatorial in nature.

Abstract

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.