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- Title
Central Limit Theorem for Mesoscopic Eigenvalue Statistics of the Free Sum of Matrices.
- Authors
Bao, Zhigang; Schnelli, Kevin; Xu, Yuanyuan
- Abstract
We consider random matrices of the form |$H_N=A_N+U_N B_N U^*_N$| , 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.
- Subjects
EIGENVALUES; CENTRAL limit theorem; HAAR integral; RANDOM matrices; STATISTICS; CHARACTERISTIC functions; MATRICES (Mathematics)
- Publication
IMRN: International Mathematics Research Notices, 2022, Vol 2022, Issue 7, p5320
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnaa210