scholarly journals Asymptotic behaviour of linear eigenvalue statistics of Hankel matrices

2021 ◽  
pp. 109273
Author(s):  
Kiran Kumar A.S. ◽  
Shambhu Nath Maurya
Keyword(s):  
Asymptotic Behaviour ◽  
Hankel Matrices ◽  
Linear Eigenvalue Statistics ◽  
Eigenvalue Statistics

Moderate deviations for linear eigenvalue statistics of β-ensembles

2021 ◽  
pp. 2250017
Author(s):  
Fuqing Gao ◽  
Jianyong Mu
Keyword(s):  
Moderate Deviation ◽  
Main Ingredient ◽  
Moderate Deviation Principle ◽  
Uniform Estimates ◽  
Linear Eigenvalue Statistics ◽  
Deviation Principle ◽  
Eigenvalue Statistics ◽  
Real Analytic ◽  
The One ◽  
Analytic Potential

We establish a moderate deviation principle for linear eigenvalue statistics of [Formula: see text]-ensembles in the one-cut regime with a real-analytic potential. The main ingredient is to obtain uniform estimates for the correlators of a family of perturbations of [Formula: see text]-ensembles using the loop equations.

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.

scholarly journals Linear eigenvalue statistics of random matrices with a variance profile

2020 ◽  
pp. 2250004
Author(s):  
Kartick Adhikari ◽  
Indrajit Jana ◽  
Koushik Saha
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Random Variable ◽  
Second Order ◽  
Linear Eigenvalue Statistics ◽  
Eigenvalue Statistics ◽  
Eigenvalue Statistic ◽  
Variation Distance ◽  
Random Matrix Ensembles ◽  
Variance Profile

We give an upper bound on the total variation distance between the linear eigenvalue statistic, properly scaled and centered, of a random matrix with a variance profile and the standard Gaussian random variable. The second-order Poincaré inequality-type result introduced in [S. Chatterjee, Fluctuations of eigenvalues and second order poincaré inequalities, Prob. Theory Rel. Fields 143(1) (2009) 1–40.] is used to establish the bound. Using this bound, we prove central limit theorem for linear eigenvalue statistics of random matrices with different kind of variance profiles. We re-establish some existing results on fluctuations of linear eigenvalue statistics of some well-known random matrix ensembles by choosing appropriate variance profiles.

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