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This is a joint work with P.Deift.

We give a rigorous 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 polynomial potentials

whose 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.