## Columbia Mathematics Department Colloquium

*``Dyson's Sine Kernel, Wigner Random Matrices, *

*and** Interacting Particle
Systems"*

### by

### Horng-Tzer Yau

### Harvard
University

Abstract:

The local eigenvalue statistics of the Gaussian Unitary
Ensemble (GUE) is given by Dyson's Sine kernel.

It was conjectured that this
law holds for a much general class of random matrices--- the universality

conjecture of random matrices. For
matrix ensembles that are unitarily invariant, there has been a great progress

using technique from orthogonal
polynomials. For the case of Hermitian Wigner random
matrices i.e. for matrix

ensembles with i.i.d.
entries are in general not unitarily invariant, the only result is due to Johansson who proved the

sine kernel for
N by N matrices that are of the form $H + t V$ where $H$ is distributed
according to a Wigner

matrix ensemble
and $V$ has the law of GUE. The
parameter $t$ is required to be of order one. Our main result

states that the Dyson's sine kernel holds for $t \ge N^{-3/4}$ i.e. for Wigner matrices with a vanishing Gaussian perturbation.

Our approach is based on
technique from interacting particle systems and key technical inputs are the
local semi-circle

law and level repulsion for
Wigner random matrices. We remark that the universality conjecture for general
Wigner matrices

could be deduced from the case $t
\ll N^{-1}$ which is still an open problem.

.

**April 29th, Wednesday, 5:00-6:00 pm**

**Mathematics 312**

**Tea will be served at 4:30pm**