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

Abstract

Let \({H_{N}=A_{N}+U_{N}B_{N}U_{N}^{\ast}}\) where AN and BN are two N-by-N Hermitian matrices and UN is a Haar-distributed random unitary matrix, and let \({\mu _{H_{N}},}\)\({\mu_{A_{N}}, \mu _{B_{N}}}\) be empirical measures of eigenvalues of matrices HN, AN, and BN, respectively. Then, it is known (see Pastur and Vasilchuk in Commun Math Phys 214:249–286, 2000) that for large N, the measure \({\mu _{H_{N}}}\) is close to the free convolution of measures \({\mu _{A_{N}}}\) and \({\mu _{B_{N}}}\) , where the free convolution is a non-linear operation on probability measures. The large deviations of the cumulative distribution function of \({\mu _{H_{N}}}\) 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 \({H_{N_{N}},}\) 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.