Bulk universality for generalized Wigner matrices

Abstract

Consider $N\times N$ Hermitian or symmetric random matrices $H$ where the distribution of the $(i,j)$ matrix element is given by a probability measure ${\nu }_{ij}$ with a subexponential decay. Let ${\sigma }_{ij}^2$ be the variance for the probability measure ${\nu }_{ij}$ with the normalization property that ${\sum }_i{\sigma }_{ij}^2=1$ for all $j$. Under essentially the only condition that $c\le N{\sigma }_{ij}^2\le c^{-1}$ for some constant $c>0$, we prove that, in the limit $N\to \infty$, the eigenvalue spacing statistics of $H$ in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth $M$ the local semicircle law holds to the energy scale $M^{-1}$.

Appendix B is simplified; an extra Assumption IV was added to Thm 6.2. On Sep 17, 2011 a small error in the conditions of Lemma 7.8 was fixed and the proof of Lemma 7.5 in pages 38-39 adjusted

References in corpus (1)

• Local semicircle law and complete delocalization for Wigner random matrices

Cited by in corpus (29)

• Universality of covariance matrices
• Random matrices: Universality of local spectral statistics of non-Hermitian matrices
• Averaging Fluctuations in Resolvents of Random Band Matrices
• Edge universality of correlation matrices
• Rigidity of Eigenvalues of Generalized Wigner Matrices
• Universality of the local regime for the block band matrices with a finite number of blocks
• Stein's method in high dimensions with applications
• Random matrices: Localization of the eigenvalues and the necessity of four moments
• Eigenvector Distribution of Wigner Matrices
• Universality for generalized Wigner matrices with Bernoulli distribution
• On the Rate of Convergence to the Marchenko--Pastur Distribution
• Random matrices: The Four Moment Theorem for Wigner ensembles
• Random matrices: Universal properties of eigenvectors
• The Isotropic Semicircle Law and Deformation of Wigner Matrices
• Random matrices: Sharp concentration of eigenvalues
• The Local Semicircle Law for a General Class of Random Matrices
• Local Circular Law for Random Matrices
• Rate of Convergence of the Expected Spectral Distribution Function to the Marchenko -- Pastur Law
• Average Density of States for Hermitian Wigner Matrices
• On the second mixed moment of the characteristic polynomials of the 1D band matrices
• Rate of Convergence of the Empirical Spectral Distribution Function to the Semi-Circular Law
• The local circular law II: the edge case
• Spectral Properties of Wigner Matrices
• A central limit theorem for the determinant of a Wigner matrix
• Universality of the second mixed moment of the characteristic polynomials of the 1D band matrices: real symmetric case
• A comment on the Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices
• The Local Circular Law III: General Case
• Tracy-Widom law for the extreme eigenvalues of sample correlation matrices
• The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices