Extremal eigenvalues of sums and products of random matrices

Abstract

In this paper, we study the limiting distribution of extremal eigenvalues of two different types of random matrices models. The first model is the sample covariance matrix model which is defined by multiplications of sample matrix and population matrix. We show that if the limiting spectral distribution of the population matrix has convex decay at the rightmost edge, then the order statistics of the population matrix determine the limiting distribution of the largest eigenvalue of the model. The second matrix model is the sum of Hermitian matrices with a Haar unitary conjugation. We prove that the law of the largest eigenvalue of the matrix weakly converges to the GUE Tracy-Widom distribution. As a result, we establish the edge universality for the model.