Local eigenvalue density for general MANOVA matrices (Q377777)

The authors study 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. They 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). The restriction to the Gaussian case, made in most earlier work on such matrices, is not made here, and the entries of \(X\) and \(Y\) are not required to be identically distributed.