Convergence Rate for Spectral Distribution of Addition of Random Matrices

Abstract

Let $A$ and $B$ be two $N$ by $N$ deterministic Hermitian matrices and let $U$ be an $N$ by $N$ Haar distributed unitary matrix. It is well known that the spectral distribution of the sum $H=A+UB{U}^{\ast }$ converges weakly to the free additive convolution of the spectral distributions of $A$ and $B$, as $N$ tends to infinity. We establish the optimal convergence rate $\frac{1}{N}$ in the bulk of the spectrum.