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