Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices
Yiting Li, Kevin Schnelli, Yuanyuan Xu
Abstract
We consider N by N deformed Wigner random matrices of the form XN=HN+AN, where HN is a real symmetric or complex Hermitian Wigner matrix and AN is a deterministic real bounded diagonal matrix. We prove a universal Central Limit Theorem for the linear eigenvalue statistics of XN for all mesoscopic scales both in the spectral bulk and at regular edges where the global eigenvalue density vanishes as a square root. The method relies on studying the characteristic function of the linear statistics (Landon and Sosoe (2018)) by using the cumulant expansion method, along with local laws for the Green function of XN (Ann. Probab. 48 (2020) 963–1001; Probab. Theory Related Fields 169 (2017) 257–352; J. Math. Phys. 54 (2013) 103504) and analytic subordination properties of the free additive convolution (Dallaporta and Fevrier (2019); Random Matrices Theory Appl. 9 (2020) 2050011). We also prove the analogous results for high-dimensional sample covariance matrices.