The semicircle law and beyond: The shape of spectra of Wigner matrices

2019 ◽  
pp. 41-73
Author(s):  
Ioana Dumitriu
Keyword(s):  
Semicircle Law ◽  
Wigner Matrices

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.

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.

Limiting Behavior of Eigenvectors of Large Wigner Matrices

2011 ◽  
Vol 146 (3) ◽  
pp. 519-549 ◽  
Author(s):  
Z. D. Bai ◽  
G. M. Pan
Keyword(s):  
Limiting Behavior ◽  
Wigner Matrices

scholarly journals Equipartition principle for Wigner matrices

2021 ◽  
Vol 9 ◽  
Author(s):  
Zhigang Bao ◽  
László Erdős ◽  
Kevin Schnelli
Keyword(s):  
High Precision ◽  
Quantum Systems ◽  
Wigner Matrices ◽  
Very High

Abstract We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices.

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