Central Limit Theorem for Mesoscopic Eigenvalue Statistics of the Free Sum of Matrices

Abstract

We consider random matrices of the form H N=A N+U N B N ∗ N, where A N and B N are two N by N deterministic Hermitian matrices and U N is a Haar distributed random unitary matrix. We establish a universal central limit theorem for the linear eigenvalue statistics of H N on all mesoscopic scales inside the regular bulk of the spectrum. The proof is based on studying the characteristic function of the linear eigenvalue statistics and consists of two main steps: (1) generating Ward identities using the left-translation invariance of the Haar measure, along with a local law for the resolvent of H N and analytic subordination properties of the free additive convolution, allows us to derive an explicit formula for the derivative of the characteristic function; (2) a local law for two-point product functions of resolvents is derived using a partial randomness decomposition of the Haar measure. We also prove the corresponding results for orthogonal conjugations.