We consider N by N deformed Wigner random matrices of the form X-N = H-N + A(N), where H-N is a real symmetric or complex Hermitian Wigner matrix and A(N) is a deterministic real bounded diagonal matrix. We prove a universal Central Limit Theorem for the linear eigenvalue statistics of X-N 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 X-N (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.

In this paper, we consider a strongly repelling model of n ordered particles {e(i theta j)}(j=0)(n-1 )with the density p(theta(0), ..., theta(n-1)) = 1/Z(n )exp { - beta/2 Sigma(j not equal k )sin(-2 )(theta(j) - theta(k)/2)}, beta > 0. Let theta(j )= 2 pi j/n + x(j)/n(2) + const such that Sigma(n-1 )(j=0)x(j )= 0. Define zeta(n) (2 pi j/n) = x(j)/root n, and extend zeta(n) piecewise linearly to [0, 2 pi]. We prove the functional convergence of zeta(n)(t) to zeta(t) = root 2/beta Re (Sigma(infinity )(k=1)1/k e(ikt) Z(k)), where Z(k ) are independent identically distributed complex standard Gaussian random variables.

We consider an N by N real or complex generalized Wigner matrix H-N, whose entries are independent centered random variables with uniformly bounded moments. We assume that the variance profile, s(ij) := E vertical bar H-ij vertical bar(2), satisfies Sigma(N)(i=1) s(ij) = 1, for all 1 <= j <= N and c(-1) <= Ns(ij) <= c for all 1 <= i, j <= N with some constant c >= 1. We establish Gaussian fluctuations for the linear eigenvalue statistics of HN on global scales, as well as on all mesoscopic scales up to the spectral edges, with the expectation and variance formulated in terms of the variance profile. We subsequently obtain the universal mesoscopic central limit theorems for the linear eigenvalue statistics inside the bulk and at the edges, respectively.

We consider random matrices of the form H-N = A(N) + UNBNUN*, 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.