Spectra of sums and products of random matrices

Abstract

We consider two different random matrix models, deformed Wigner matrices and products of unitarily invariant positive definite matrices. For deformed Wigner matrices, that are defined by the sum of Wigner and diagonal matrices, we prove that their linear eigenvalue statistics have asymptotically Gaussian fluctuation as the size of matrices diverges. For the second model, that is the product of two unitarily invariant matrices, we prove an optimal local law around the spectral edge. As its application, we prove eigenvalue rigidity and eigenvector delocalization around the edge.