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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 UBU{*} 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 1/N in the bulk of the spectrum. (C) 2017 Elsevier Inc. All rights reserved.

We consider the free additive convolution of two probability measures, supported on respectively n and m disjoint bounded intervals on the real line, and derive a lower bound and an upper bound that is strictly smaller than 2nm, on the number of connected components in its support. We also obtain the corresponding results for the free additive convolution semi-group. Throughout the paper,we consider classes of probability measures with power law behaviors at the endpoints of their supports with exponents ranging from −1 to 1. Our main theorem generalizes a result of Bao, Erdös and Schnelli to the multi-cut setup., QC 20221005