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Lectures on the local semicircle law for Wigner matrices
TLDR
The local semicircle law of Wigner matrices has been studied in this paper, which states that the eigenvalue distribution of a random matrix with independent upper-triangular entries with zero expectation and constant variance is very similar to Wigners' distribution, down to spectral scales containing slightly more than one eigen value.Abstract:
These notes provide an introduction to the local semicircle law from random matrix theory, as well as some of its applications. We focus on Wigner matrices, Hermitian random matrices with independent upper-triangular entries with zero expectation and constant variance. We state and prove the local semicircle law, which says that the eigenvalue distribution of a Wigner matrix is close to Wigner's semicircle distribution, down to spectral scales containing slightly more than one eigenvalue. This local semicircle law is formulated using the Green function, whose individual entries are controlled by large deviation bounds.
We then discuss three applications of the local semicircle law: first, complete delocalization of the eigenvectors, stating that with high probability the eigenvectors are approximately flat; second, rigidity of the eigenvalues, giving large deviation bounds on the locations of the individual eigenvalues; third, a comparison argument for the local eigenvalue statistics in the bulk spectrum, showing that the local eigenvalue statistics of two Wigner matrices coincide provided the first four moments of their entries coincide. We also sketch further applications to eigenvalues near the spectral edge, and to the distribution of eigenvectors.read more
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References
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Book
Spectral Analysis of Large Dimensional Random Matrices
Zhidong Bai,Jack W. Silverstein +1 more
TL;DR: Wigner Matrices and Semicircular Law for Hadamard products have been used in this article for spectral separations and convergence rates of ESD for linear spectral statistics.
Journal ArticleDOI
On the Distribution of the Roots of Certain Symmetric Matrices
TL;DR: The distribution law obtained before' for a very special set of matrices is valid for much more general sets of real symmetric matrices of very high dimensionality.
Book
An Introduction to Random Matrices
TL;DR: The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial) as mentioned in this paper.
Journal ArticleDOI
A Brownian‐Motion Model for the Eigenvalues of a Random Matrix
TL;DR: In this paper, a new type of Coulomb gas is defined, consisting of n point charges executing Brownian motions under the influence of their mutual electrostatic repulsions, and it is proved that this gas gives an exact mathematical description of the behavior of the eigenvalues of an (n × n) Hermitian matrix, when the elements of the matrix execute independent Brownian motion without mutual interaction.