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

Let H_N = A_N + U_NB_NU_N^*where A_N and B_N are two N-by-N Hermitian matrices and U_N is a Haar-distributed random unitary matrix, and let μH_N, μA_N, μ B_N be empirical measures of eigenvalues of matrices H_N, A_N, and B_N, respectively. Then, it is known (see Pastur and Vasilchuk in Commun Math Phys 214:249-286, 2000) that for large N, the measure μ H_N is close to the free convolution of measures μA_N and μB_N where the free convolution is a non-linear operation on probability measures. The large deviations of the cumulative distribution function of μH_N from its expectation have been studied by Chatterjee (J Funct Anal 245:379-389, 2007). 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 HN_N, by showing that the normalized number of eigenvalues in an interval approaches the density of the free convolution of μ _A and μ _B provided that the interval has width (log N)^-1/2.