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

Let 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 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 is close to the free convolution of measures and , where the free convolution is a non-linear operation on probability measures. The large deviations of the cumulative distribution function of 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 by showing that the normalized number of eigenvalues in an interval approaches the density of the free convolution of mu (A) and mu (B) provided that the interval has width (log N)(-1/2).