The spectra of random abelian G-circulant matrices

Abstract

This paper studies the asymptotic behavior of eigenvalues of random abelian G-circulant matrices, that is, matrices whose structure is related to a finite abelian group G in a way that naturally generalizes the relationship between circulant matrices and cyclic groups. It is shown that, under mild conditions, when the size of the group G goes to infinity, the spectral measures of such random matrices approach a deterministic limit. Depending on some aspects of the structure of the groups, whether the matrices are constrained to be Hermitian, and a few details of the distributions of the matrix entries, the limit measure is either a (complex or real) Gaussian distribution or a mixture of two Gaussian distributions.