### 11 oktober,
Dimitri Gioev,
Universality in Random Matrix Theory
for Orthogonal and Symplectic Ensembles

**Abstract.**
We give a proof of the Universality Conjecture in Random Matrix Theory
for orthogonal (beta=1) and symplectic (beta=4) ensembles in the scaling
limit for a class of weights w(x)=exp(-V(x)) on the line
where V(x) is a polynomial.
(For such weights the associated equilibrium measure is supported on a
single interval.)
Our starting point is Widom's representation of the correlation kernels
for the beta=1,4 cases in terms of the unitary (beta=2) correlation
kernel
plus a correction.
In the asymptotic analysis of the correction terms
we use amongst other things differential equations for the derivatives
of
orthogonal polynomials (OP's) due to Tracy-Widom,
and uniform Plancherel-Rotach type asymptotics for OP's
due to Deift-Kriecherbauer-McLaughlin-Venakides-Zhou.
The problem reduces to a small norm problem for a certain matrix of a
fixed size that is equal to the degree of the polynomial potential.
This is a joint work with Percy Deift (Courant Institute).