On Wigner's semicircle law for the eigenvalues of random matrices

1971 ◽  
Vol 19 (3) ◽  
pp. 191-198 ◽  
Author(s):  
L. Arnold
Keyword(s):  
Random Matrices ◽  
Semicircle Law ◽  
Eigenvalues Of Random Matrices

scholarly journals Small-Deviation Inequalities for Sums of Random Matrices

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 638
Author(s):  
Xianjie Gao ◽  
Chao Zhang ◽  
Hongwei Zhang
Keyword(s):  
Random Matrices ◽  
Large Deviation ◽  
Small Deviation ◽  
High Dimensional ◽  
Main Research ◽  
Infinite Dimensional ◽  
The Matrix ◽  
Largest Eigenvalues ◽  
Deviation Inequalities ◽  
Eigenvalues Of Random Matrices

Random matrices have played an important role in many fields including machine learning, quantum information theory, and optimization. One of the main research focuses is on the deviation inequalities for eigenvalues of random matrices. Although there are intensive studies on the large-deviation inequalities for random matrices, only a few works discuss the small-deviation behavior of random matrices. In this paper, we present the small-deviation inequalities for the largest eigenvalues of sums of random matrices. Since the resulting inequalities are independent of the matrix dimension, they are applicable to high-dimensional and even the infinite-dimensional cases.

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 Random Matrices From Linear Codes and Wigner’s Semicircle Law

2019 ◽  
Vol 65 (10) ◽  
pp. 6001-6009
Author(s):  
Chin Hei Chan ◽  
Enoch Kung ◽  
Maosheng Xiong
Keyword(s):  
Random Matrices ◽  
Linear Codes ◽  
Semicircle Law

The statistics of the eigenvalues of random matrices

1966 ◽  
Vol 78 (3) ◽  
pp. 553-556 ◽  
Author(s):  
F. Cristofori ◽  
P.G. Sona ◽  
F. Tonolini
Keyword(s):  
Random Matrices ◽  
Eigenvalues Of Random Matrices
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