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We consider random matrices of the form $H_N=A_N+U_N B_N U^*_N$, where $A_N$,$B_N$ are two $N$ by $N$ deterministic Hermitian matrices and $U_N$ is a Haardistributed random unitary matrix. We establish a universal Central LimitTheorem for the linear eigenvalue statistics of $H_N$ on all mesoscopic scalesinside the regular bulk of the spectrum. The proof is based on studying thecharacteristic function of the linear eigenvalue statistics, and consists oftwo main steps: (1) generating Ward identities using theleft-translation-invariance of the Haar measure, along with a local law for theresolvent of $H_N$ and analytic subordination properties of the free additiveconvolution, allow us to derive an explicit formula for the derivative of thecharacteristic function; (2) a local law for two-point product functions ofresolvents is derived using a partial randomness decomposition of the Haarmeasure. We also prove the corresponding results for orthogonal conjugations.

Central limit theorem for mesoscopic eigenvalue statistics of the free sum of matrices