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

Abstract

We consider random matrices of the form $H_N=A_N+U_N B_N U^*_N$, 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.

Fluctuation around the circular law for random matrices with real entries

“…As in the complex case [22], one key ingredient for both Propositions 3.3 and 3.4 is a local law for products of resolvents G 1 , G 2 for G i = G zi (w i ). We remark that local laws for products of resolvents have also been derived for (generalized) Wigner matrices [30,46] and for sample covariance matrices [20], as well as for the addition of random matrices [8].…”

On fluctuations of global and mesoscopic linear statistics of generalized Wigner matrices

“…Compared with the standard Wigner matrices [42,53], the two point function T ab (z, z ) cannot be written as a matrix product and hence the resolvent identity (5.15) or cyclicity of trace no longer help. Similar two point functions of the resolvents appeared in [34,22,24,10] to derive Gaussian fluctuations of the linear eigenvalue statistics for different random matrix ensembles. The proof of Lemma 4.3 is inspired by the fluctuation averaging mechanism [29], combined with recursive moment estimates based on cumulant expansions.…”

“…Besides Wigner matrices, mesoscopic CLTs were also obtained in many other random matrices ensembles, e.g., random band matrices [27,28], sparse Wigner matrices [41], Dyson Brownian motion [26,47,54], invariant β-ensembles [12,15,51], orthogonal polynomial ensembles [20], classical compact groups [66], circular β ensembles [52], and free sum of matrices [10].…”

Global and Local Scaling Limits for Linear Eigenvalue Statistics of Jacobi $β$-Ensembles

“…Linear statistics is an important research object in RMT and has various applications; see, e.g., [2,[4][5][6][7][8]12,15,17,25].…”