scholarly journals Central Limit Theorem for Linear Eigenvalue Statistics of Elliptic Random Matrices

2015 ◽  
Vol 29 (3) ◽  
pp. 1121-1191 ◽  
Author(s):  
Sean O’Rourke ◽  
David Renfrew
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Matrices ◽  
Central Limit ◽  
Linear Eigenvalue Statistics ◽  
Eigenvalue Statistics

On the fluctuations of eigenvalues of multiplicative deformed unitary invariant ensembles

2016 ◽  
Vol 05 (02) ◽  
pp. 1650007 ◽  
Author(s):  
Vladimir Vasilchuk
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Matrices ◽  
Central Limit ◽  
Counting Measure ◽  
Linear Eigenvalue Statistics ◽  
The Central Limit Theorem ◽  
Eigenvalue Statistics ◽  
Leading Term

We consider the ensemble of [Formula: see text] random matrices [Formula: see text], where [Formula: see text] and [Formula: see text] are non-random, unitary, having the limiting Normalized Counting Measure (NCM) of eigenvalues, and [Formula: see text] is unitary, uniformly distributed over [Formula: see text]. We find the leading term of the covariance of traces of resolvent of [Formula: see text] and establish the Central Limit Theorem for sufficiently smooth linear eigenvalue statistics of [Formula: see text] as [Formula: see text].

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

2020 ◽  
Author(s):  
Zhigang Bao ◽  
Kevin Schnelli ◽  
Yuanyuan Xu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Characteristic Function ◽  
Central Limit ◽  
Haar Measure ◽  
Unitary Matrix ◽  
Translation Invariance ◽  
Linear Eigenvalue Statistics ◽  
Eigenvalue Statistics ◽  
Local Law

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.

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