scholarly journals Gaussian fluctuations for linear spectral statistics of deformed Wigner matrices

2019 ◽  
Vol 09 (03) ◽  
pp. 2050011
Author(s):  
Hong Chang Ji ◽  
Ji Oon Lee
Keyword(s):  
Large Class ◽  
Random Matrices ◽  
Gaussian Distribution ◽  
Diagonal Matrix ◽  
Limiting Distributions ◽  
Test Function ◽  
Wigner Matrix ◽  
Wigner Matrices ◽  
Gaussian Fluctuations ◽  
Spectral Statistics

We consider large-dimensional Hermitian or symmetric random matrices of the form [Formula: see text], where [Formula: see text] is a Wigner matrix and [Formula: see text] is a real diagonal matrix whose entries are independent of [Formula: see text]. For a large class of diagonal matrices [Formula: see text], we prove that the fluctuations of linear spectral statistics of [Formula: see text] for [Formula: see text] test function can be decomposed into that of [Formula: see text] and of [Formula: see text], and that each of those weakly converges to a Gaussian distribution. We also calculate the formulae for the means and variances of the limiting distributions.

scholarly journals Edge universality for deformed Wigner matrices

2015 ◽  
Vol 27 (08) ◽  
pp. 1550018 ◽  
Author(s):  
Ji Oon Lee ◽  
Kevin Schnelli
Keyword(s):  
Large Class ◽  
Random Matrices ◽  
Diagonal Matrix ◽  
Wigner Matrix ◽  
Wigner Matrices ◽  
Large N ◽  
The Matrix

We consider N × N random matrices of the form H = W + V where W is a real symmetric Wigner matrix and V a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the rescaled distribution of the extremal eigenvalues is given by the Tracy–Widom distribution F1 in the limit of large N. Our proofs also apply to the complex Hermitian setting, i.e. when W is a complex Hermitian Wigner matrix.

scholarly journals Nonuniversality of fluctuations of outliers for Hermitian polynomials in a complex Wigner matrix and a spiked diagonal matrix

2019 ◽  
Vol 09 (04) ◽  
pp. 2050013
Author(s):  
Mireille Capitaine
Keyword(s):  
Probability Theory ◽  
Free Probability ◽  
Diagonal Matrix ◽  
Free Probability Theory ◽  
Wigner Matrix ◽  
Wigner Matrices ◽  
Hermitian Polynomial ◽  
Hermitian Polynomials

We study the fluctuations associated to the a.s. convergence of the outliers established by Belinschi–Bercovici–Capitaine of an Hermitian polynomial in a complex Wigner matrix and a spiked deterministic real diagonal matrix. Thus, we extend the nonuniversality phenomenon established by Capitaine–Donati-Martin–Féral for additive deformations of complex Wigner matrices, to any Hermitian polynomial. The result is described using the operator-valued subordination functions of free probability theory.

scholarly journals On the Outlying Eigenvalues of a Polynomial in Large Independent Random Matrices

2019 ◽  
Author(s):  
Serban T Belinschi ◽  
Hari Bercovici ◽  
Mireille Capitaine
Keyword(s):  
Probability Theory ◽  
Random Matrices ◽  
Empirical Distribution ◽  
Free Probability ◽  
Probability Measures ◽  
Unitary Operators ◽  
Free Probability Theory ◽  
Wigner Matrix ◽  
Asymptotic Eigenvalue ◽  
Special Case

Abstract Given a selfadjoint polynomial $P(X,Y)$ in two noncommuting selfadjoint indeterminates, we investigate the asymptotic eigenvalue behavior of the random matrix $P(A_N,B_N)$, where $A_N$ and $B_N$ are independent Hermitian random matrices and the distribution of $B_N$ is invariant under conjugation by unitary operators. We assume that the empirical eigenvalue distributions of $A_N$ and $B_N$ converge almost surely to deterministic probability measures $\mu$ and $\nu$, respectively. In addition, the eigenvalues of $A_N$ and $B_N$ are assumed to converge uniformly almost surely to the support of $\mu$ and $\nu ,$ respectively, except for a fixed finite number of fixed eigenvalues (spikes) of $A_N$. It is known that almost surely the empirical distribution of the eigenvalues of $P(A_N,B_N)$ converges to a certain deterministic probability measure $\eta \ (\textrm{sometimes denoted}\ P^\square(\mu,\nu))$ and, when there are no spikes, the eigenvalues of $P(A_N,B_N)$ converge uniformly almost surely to the support of $\eta$. When spikes are present, we show that the eigenvalues of $P(A_N,B_N)$ still converge uniformly to the support of $\eta$, with the possible exception of certain isolated outliers whose location can be determined in terms of $\mu ,\nu ,P$, and the spikes of $A_N$. We establish a similar result when $B_N$ is replaced by a Wigner matrix. The relation between outliers and spikes is described using the operator-valued subordination functions of free probability theory. These results extend known facts from the special case in which $P(X,Y)=X+Y$.

scholarly journals ON FINITE RANK DEFORMATIONS OF WIGNER MATRICES II: DELOCALIZED PERTURBATIONS

2013 ◽  
Vol 02 (01) ◽  
pp. 1250015 ◽  
Author(s):  
DAVID RENFREW ◽  
ALEXANDER SOSHNIKOV
Keyword(s):  
Random Matrices ◽  
Finite Rank ◽  
Fourth Moment ◽  
Wigner Matrices ◽  
Wigner Random Matrices ◽  
The Matrix

We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrices. We assume that the matrix entries have finite fourth moment and extend the results by Capitaine, Donati-Martin, and Féral for perturbations whose eigenvectors are delocalized.

Pair dependent linear statistics for CβE

2020 ◽  
pp. 2150035
Author(s):  
Ander Aguirre ◽  
Alexander Soshnikov ◽  
Joshua Sumpter
Keyword(s):  
Zeta Function ◽  
Random Matrices ◽  
Riemann Zeta Function ◽  
Classical Result ◽  
Pair Correlation ◽  
Test Function ◽  
Linear Statistics ◽  
Smooth Test ◽  
Riemann Zeta ◽  
Counting Statistics

We study the limiting distribution of a pair counting statistics of the form [Formula: see text] for the circular [Formula: see text]-ensemble (C[Formula: see text]E) of random matrices for sufficiently smooth test function [Formula: see text] and [Formula: see text] For [Formula: see text] and [Formula: see text] our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.

scholarly journals EIGENVALUE VARIANCE BOUNDS FOR WIGNER AND COVARIANCE RANDOM MATRICES

2012 ◽  
Vol 01 (03) ◽  
pp. 1250007 ◽  
Author(s):  
S. DALLAPORTA
Keyword(s):  
Random Matrices ◽  
Spectral Measure ◽  
Wasserstein Distance ◽  
Covariance Matrices ◽  
Finite Range ◽  
Variance Bounds ◽  
Semicircle Law ◽  
Wigner Matrices ◽  
Wigner Random Matrices

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erdös, Yau and Yin. The case of real Wigner matrices is obtained from interlacing formulas. As an application, bounds on the expected 2-Wasserstein distance between the empirical spectral measure and the semicircle law are derived. Similar results are available for random covariance matrices.

scholarly journals Asymptotic Linear Spectral Statistics for Spiked Hermitian Random Matrices

2015 ◽  
Vol 160 (1) ◽  
pp. 120-150 ◽  
Author(s):  
Damien Passemier ◽  
Matthew R. McKay ◽  
Yang Chen
Keyword(s):  
Random Matrices ◽  
Spectral Statistics
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