Rigidity of eigenvalues of generalized Wigner matrices

@article{Erds2010RigidityOE,
  title={Rigidity of eigenvalues of generalized Wigner matrices},
  author={L. Erdős and H. Yau and Jun Yin},
  journal={Advances in Mathematics},
  year={2010},
  volume={229},
  pages={1435-1515}
}
Consider N×N Hermitian or symmetric random matrices H with independent entries, where the distribution of the (i,j) matrix element is given by the probability measure νij with zero expectation and with variance σij2. We assume that the variances satisfy the normalization condition ∑iσij2=1 for all j and that there is a positive constant c such that c⩽Nσij2⩽c−1. We further assume that the probability distributions νij have a uniform subexponential decay. We prove that the Stieltjes transform of… Expand
Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices
We consider random matrices of the form H = W + λV , λ ∈ R, where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of size N withExpand
Universality of Wigner random matrices: a survey of recent results
This is a study of the universality of spectral statistics for large random matrices. Considered are symmetric, Hermitian, or quaternion self-dual random matrices with independent identicallyExpand
Random covariance matrices: Universality of local statistics of eigenvalues
We study the eigenvalues of the covariance matrix 1/n M∗M of a large rectangular matrix M = Mn,p = (ζij)1≤i≤p;1≤j≤n whose entries are i.i.d. random variables of mean zero, variance one, and havingExpand
Eigenvector distribution of Wigner matrices
We consider N × N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)-th matrix element is given by a probability measure νij whose first two momentsExpand
Universality of random matrices and local relaxation flow
Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, weExpand
Wigner matrices with random potential
Consider large matrices whose entries are random variables. Famous examples of such matrices are Wigner matrices: a Wigner matrix is an N × N real or complex matrix W = (wij) whose entries areExpand
Lectures on the local semicircle law for Wigner matrices
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 independentExpand
Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices
Abstract We show that the variance of centred linear statistics of eigenvalues of GUE matrices remains bounded for large n for some classes of test functions less regular than Lipschitz functions.Expand
Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues
We prove a new CLT for the difference of linear eigenvalue statistics of a Wigner random matrix H and its minor Ĥ and find that the fluctuation is much smaller than the fluctuations of the individualExpand
The Isotropic Semicircle Law and Deformation of Wigner Matrices
We analyze the spectrum of additive finite-rank deformations of N × N Wigner matrices H. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of anExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 56 REFERENCES
Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices
AbstractWe study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble.We begin by considering an n×n matrix from the GaussianExpand
Local Semicircle Law and Complete Delocalization for Wigner Random Matrices
We consider N × N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. UnderExpand
Universality of random matrices and local relaxation flow
Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, weExpand
Wegner estimate and level repulsion for Wigner random matrices
We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutiveExpand
Universality for generalized Wigner matrices with Bernoulli distribution
The universality for the eigenvalue spacing statistics of generalized Wigner matrices was established in our previous work \cite{EYY} under certain conditions on the probability distributions of theExpand
Bulk universality for generalized Wigner matrices
Consider N × N Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure νij with a subexponential decay. Let $${\sigma_{ij}^2}$$Expand
Introduction to Random Matrices
Here I = S j (a2j 1,a2j) andI(y) is the characteristic function of the set I. In the Gaussian Unitary Ensemble (GUE) the probability that no eigenvalues lie in I is equal to �(a). Also �(a) is aExpand
Universality of the Edge Distribution of Eigenvalues of Wigner Random Matrices with Polynomially Decaying Distributions of Entries
AbstractWe consider an ensemble of Wigner symmetric random matrices An={aij}, i,j=1, . . . ,n with matrix elements aij, being i.i.d. symmetrically distributed random variables We assume that and thatExpand
Bulk Universality for Wigner Matrices
We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e−U(x). We prove that the eigenvalue statistics in the bulkExpand
Universality of local eigenvalue statistics for some sample covariance matrices
We consider random, complex sample covariance matrices 1 X ∗ X, where X is a p× N random matrix with i.i.d. entries of distribution � . It has been conjectured that both the distribution of theExpand
...
1
2
3
4
5
...