  We give a proof of the Universality Conjecture for orthogonal and symplectic
ensembles of random matrices in the scaling limit for a class of weights
w(x)=exp(-V(x)) where V is a polynomial, V(x)=kappa_{2m}x^{2m}+...,
kappa_{2m}>0. For such weights the associated equilibrium measure is supported
on a single interval. The precise statement of our results is given in Theorem
1.1 below. For a proof of the Universality Conjecture for unitary ensembles,
for the same class of weights, see [DKMVZ2].
  Our starting point is Widom's representation [W] of the orthogonal and
symplectic correlation kernels in terms of the kernel arising in the unitary
case plus a correction term which is constructed out of derivatives and
integrals of orthonormal polynomials (OP's) {p_j(x)}, j=0,1,..., with respect
to the weight w(x). The calculations in [W] in turn depend on the earlier work
of Tracy and Widom [TW2]. It turns out (see [W] and also Theorems 2.1 and 2.2
below) that only the OP's in the range j=N+O(1), N->infinity, contribute to the
correction term. In controlling this correction term, and hence proving
Universality for both the orthogonal and symplectic cases, the uniform
Plancherel--Rotach type asymptotics for the OP's found in [DKMVZ2] play an
important role, but there are significant new analytical difficulties that must
be overcome which are not present in the unitary case. We note that we do not
use skew orthogonal polynomials.
