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Rigidity of eigenvalues of generalized Wigner matrices

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… Expand

Anisotropic local laws for random matrices

- A. Knowles, Jun Yin
- Mathematics, Physics
- 13 October 2014

We develop a new method for deriving local laws for a large class of random matrices. It is applicable to many matrix models built from sums and products of deterministic or independent random… Expand

Spectral statistics of Erdős–Rényi graphs I: Local semicircle law

- L'aszl'o ErdHos, A. Knowles, H. Yau, Jun Yin
- Mathematics, Physics
- 9 March 2011

We consider the ensemble of adjacency matrices of Erdős–Renyi random graphs, that is, graphs on N vertices where every edge is chosen independently and with probability p≡p(N). We rescale the matrix… Expand

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

The local semicircle law for a general class of random matrices

- L. Erdős, A. Knowles, H. Yau, Jun Yin
- Mathematics, Physics
- 1 December 2012

We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local… Expand

Isotropic local laws for sample covariance and generalized Wigner matrices

- Alex Bloemendal, L. Erdős, A. Knowles, H. Yau, Jun Yin
- Mathematics, Physics
- 27 August 2013

We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M \times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that… Expand

Universality of covariance matrices

In this paper we prove the universality of covariance matrices of the form $H_{N\times N}={X}^{\dagger}X$ where $X$ is an ${M\times N}$ rectangular matrix with independent real valued entries… Expand

Fixed energy universality for generalized Wigner matrices

- P. Bourgade, L. Erdős, Hong-Tzer Yau, Jun Yin
- Mathematics, Physics
- 21 July 2014

We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random… Expand

Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

- L. Erdős, A. Knowles, H. Yau, Jun Yin
- Mathematics, Physics
- 20 March 2011

We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so… Expand

Eigenvector distribution of Wigner matrices

- A. Knowles, Jun Yin
- Mathematics, Physics
- 1 February 2011

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 moments… Expand

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