Universality and sharp matrix concentration inequalities

Abstract

We show that, under mild assumptions, the spectrum of a sum of independent random matrices is close to that of the Gaussian random matrix whose entries have the same mean and covariance. This nonasymptotic universality principle both explains the phenomenon behind classical matrix concentration inequalities such as the matrix Bernstein inequality, and yields new sharp matrix concentration inequalities for general sums of independent random matrices when combined with the recent Gaussian theory of Bandeira, Boedihardjo, and Van Handel. As an application of our main results, we prove strong asymptotic freeness of a very general class of random matrix models with non-Gaussian, nonhomogeneous, and dependent entries.Comment: 56 page