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Abstract Abstract
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21.1 Introduction 21.1 Introduction
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21.2 Global properties 21.2 Global properties
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21.2.3 Deviations and concentration properties 21.2.3 Deviations and concentration properties
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21.2.4 The local semicircle law and delocalization of the eigenvectors 21.2.4 The local semicircle law and delocalization of the eigenvectors
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21.2.5 The limits of universality: the spectral measure of heavy-tailed Wigner random matrices 21.2.5 The limits of universality: the spectral measure of heavy-tailed Wigner random matrices
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21.3 Local properties in the bulk 21.3 Local properties in the bulk
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21.3.1 Spacings in the bulk for Gaussian Wigner ensembles 21.3.1 Spacings in the bulk for Gaussian Wigner ensembles
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21.3.2 Universality for Gaussian divisible Wigner matrices 21.3.2 Universality for Gaussian divisible Wigner matrices
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21.3.3 Universality and the four moments theorem 21.3.3 Universality and the four moments theorem
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21.3.4 The extreme gaps and fluctuations of eigenvalues in the bulk 21.3.4 The extreme gaps and fluctuations of eigenvalues in the bulk
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21.4 Local properties at the edge 21.4 Local properties at the edge
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21.4.1 Convergence of the extreme eigenvalues 21.4.1 Convergence of the extreme eigenvalues
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21.4.2 Fluctuations of extreme eigenvalues and the Tracy–Widom law 21.4.2 Fluctuations of extreme eigenvalues and the Tracy–Widom law
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21.4.3 Universality 21.4.3 Universality
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21.4.4 Universality and the four moments theorem 21.4.4 Universality and the four moments theorem
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21.4.5 Extreme eigenvalues of heavy-tailed Wigner matrices 21.4.5 Extreme eigenvalues of heavy-tailed Wigner matrices
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21.4.6 Non-universality and extreme eigenvalues of finite rank perturbation of Wigner matrices 21.4.6 Non-universality and extreme eigenvalues of finite rank perturbation of Wigner matrices
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Acknowledgements Acknowledgements
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References References
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21 Wigner matrices
Get accessGérard Ben Arous, Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, Room 702, 251 Mercer St., New York, NY 10012, USA, [email protected]
Alice Guionnet, Unité de Mathématiques Pures et Appliquées, École Normale Supérieure, Lyon 69007, France, [email protected]
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Published:08 August 2018
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Abstract
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.
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