Local Eigenvalue Density for General MANOVA Matrices

TL;DR

This paper establishes that the eigenvalue density of general MANOVA matrices converges locally to the Jacobi ensemble density on the smallest possible scales, extending classical results to a broader class of matrices.

Contribution

It proves a local eigenvalue density convergence for general MANOVA matrices, generalizing known results from Gaussian to non-Gaussian entries.

Findings

Eigenvalue density converges to Jacobi ensemble density away from spectral edges.

Convergence occurs on scales of order 1/n, up to logarithmic factors.

Results extend local spectral laws to more general MANOVA matrices.

Abstract

We consider random n\times n matrices of the form (XX*+YY*)^{-1/2}YY*(XX*+YY*)^{-1/2}, where X and Y have independent entries with zero mean and variance one. These matrices are the natural generalization of the Gaussian case, which are known as MANOVA matrices and which have joint eigenvalue density given by the third classical ensemble, the Jacobi ensemble. We show that, away from the spectral edge, the eigenvalue density converges to the limiting density of the Jacobi ensemble even on the shortest possible scales of order 1/n (up to \log n factors). This result is the analogue of the local Wigner semicircle law and the local Marchenko-Pastur law for general MANOVA matrices.