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A concentration inequality and a local law for the sum of two random matrices

Let H=A+UBU* where A and B are two N-by-N Hermitian matrices and U is a Haar-distributed random unitary matrix, and let μ_H, μ_A, and μ_B be empirical measures of eigenvalues of matrices H, A, and B, respectively. Then, it is known (see, for example, Pastur-Vasilchuk, CMP, 2000, v.214, pp.249-286) that for large N, measure μ_H is close to the free convolution of measures μ_A and μ_B, where the free convolution is a non-linear operation on probability measures. The large deviations of the cumulative distribution function of μ_H from its expectation have been studied by Chatterjee in in JFA, 2007, v. 245, pp.379-389. In this paper we improve Chatterjee's concentration inequality and show that it holds with the rate which is quadratic in N. In addition, we prove a local law for eigenvalues of H, by showing that the normalized number of eigenvalues in an interval converges to the density of the free convolution of μ_A and μ_B provided that the interval has width (log N)^{-1/2}.