We prove universality at the edge of the spectrum for unitary (beta=2),
orthogonal (beta=1) and symplectic (beta=4) 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. The precise statement of our results
is given in Theorem 1.1 and Corollaries 1.2, 1.3 below. For a proof of
universality in the bulk of the spectrum, for the same class of weights, for
unitary ensembles see [DKMVZ2], and for orthogonal and symplectic ensembles see
[DG].
Our starting point in the unitary case is [DKMVZ2], and for the orthogonal
and symplectic cases we rely on our recent work [DG], which in turn depends on
the earlier work of Widom [W] and Tracy and Widom [TW2]. As in [DG], the
uniform Plancherel--Rotach type asymptotics for the orthogonal polynomials
found in [DKMVZ2] plays a central role.
The formulae in [W] express the correlation kernels for beta=1 and 4 as a sum
of a Christoffel--Darboux (CD) term, as in the case beta=2, together with a
correction term. In the bulk scaling limit [DG], the correction term is of
lower order and does not contribute to the limiting form of the correlation
kernel. By contrast, in the edge scaling limit considered here, the CD term and
the correction term contribute to the same order: this leads to additional
technical difficulties over and above [DG].