The Limiting Spectral Measure for the Ensemble of Generalized Real Symmetric Block m-Circulant Matrices

Given an ensemble of N × N random matrices with independent entries chosen from a nice probability distribution, a natural question is whether the empirical spectral measures of typical matrices converge to some limiting measure as N → ∞. It has been shown that the limiting spectral distribution for the ensemble of real symmetric matrices is a semi-circle, and that the distribution for real symmetric circulant matrices is a Gaussian. As a transition from the general real symmetric matrices to the highly structured circulant matrices, the ensemble of block m-circulant matrices with toroidal diagonals of period m exhibits an eigenvalue density as the product of a Gaussian and a certain even polynomial of degree 2m − 2. This paper generalizes the m-circulant pattern and shows that the limiting spectral distribution is determined by the pattern of the elements within an m-period, depending on not only the frequency with which each element appears, but also the way the elements are arranged. For an arbitrary pattern, the empirical spectral measures converge to some nice probability distribution as N →∞.