Columbia Mathematics Department Colloquium


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

and Interacting Particle Systems"


 Horng-Tzer Yau  

Harvard University



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