Wigner’s semicircle law for band random matrices

1993 ◽  
Vol 1 (1) ◽  
Author(s):  
G. CASATI ◽  
V. GIRKO
Keyword(s):  
Random Matrices ◽  
Semicircle Law

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

Wigner matrices

2018 ◽  
Author(s):  
Peter Forrester
Keyword(s):  
Simple Model ◽  
Random Matrices ◽  
Symmetric Matrices ◽  
Semicircle Law ◽  
Wigner Matrices ◽  
Global Properties ◽  
Wigner Random Matrices ◽  
Local Properties ◽  
Concentration Properties ◽  
Adjoint Operators

This article reviews some of the important results in the study of the eigenvalues and the eigenvectors of Wigner random matrices, that is. random Hermitian (or real symmetric) matrices with iid entries. It first provides an overview of the Wigner matrices, introduced in the 1950s by Wigner as a very simple model of random matrices to approximate generic self-adjoint operators. It then considers the global properties of the spectrum of Wigner matrices, focusing on convergence to the semicircle law, fluctuations around the semicircle law, deviations and concentration properties, and the delocalization of the eigenvectors. It also describes local properties in the bulk and at the edge before concluding with a brief analysis of the known universality results showing how much the behaviour of the spectrum is insensitive to the distribution of the entries.

scholarly journals Local Semicircle Law and Complete Delocalization for Wigner Random Matrices

2008 ◽  
Vol 287 (2) ◽  
pp. 641-655 ◽  
Author(s):  
László Erdős ◽  
Benjamin Schlein ◽  
Horng-Tzer Yau
Keyword(s):  
Random Matrices ◽  
Semicircle Law ◽  
Local Semicircle Law ◽  
Wigner Random Matrices

scholarly journals On the universality of spectral limit for random matrices with martingale differences entries

2015 ◽  
Vol 04 (01) ◽  
pp. 1550003 ◽  
Author(s):  
F. Merlevède ◽  
C. Peligrad ◽  
M. Peligrad
Keyword(s):  
Random Matrices ◽  
Infinite Order ◽  
Regularity Conditions ◽  
Martingale Differences ◽  
Full Extension ◽  
Semicircle Law ◽  
Convergence Results ◽  
Nonlinear Autoregressive ◽  
Martingale Case ◽  
Variance Structure

For a class of symmetric random matrices whose entries are martingale differences adapted to an increasing filtration, we prove that under a Lindeberg-like condition, the empirical spectral distribution behaves asymptotically similarly to a corresponding matrix with independent centered Gaussian entries having the same variances. Under a slightly reinforced condition, the approximation holds in the almost sure sense. We also point out several sufficient regularity conditions imposed to the variance structure for convergence to the semicircle law or the Marchenko–Pastur law and other convergence results. In the stationary case, we obtain a full extension from the i.i.d. case to the martingale case of the convergence to the semicircle law as well as to the Marchenko–Pastur one. Our results are well adapted to study several examples including nonlinear autoregressive conditional heteroscedastic random fields of infinite order.

scholarly journals The local semicircle law for a general class of random matrices

2013 ◽  
Vol 18 (0) ◽  
Author(s):  
László Erdős ◽  
Antti Knowles ◽  
Horng-Tzer Yau ◽  
Jun Yin
Keyword(s):  
Random Matrices ◽  
General Class ◽  
Semicircle Law ◽  
Local Semicircle Law

scholarly journals Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices

2009 ◽  
Vol 37 (3) ◽  
pp. 815-852 ◽  
Author(s):  
László Erdős ◽  
Benjamin Schlein ◽  
Horng-Tzer Yau
Keyword(s):  
Random Matrices ◽  
Semicircle Law ◽  
Wigner Random Matrices
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