Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices

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) = e(-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 1.3 and 1.4 below. For the same class of weights, a proof of universality in the bulk of the spectrum is given in [12] for the unitary ensembles and in [9] for the orthogonal and symplectic ensembles. Our starting point in the unitary case is [121, and for the orthogonal and symplectic cases we rely on our recent work [9], which in turn depends on the earlier work of Widom [46] and Tracy and Widom [42]. As in [9], the uniform Plancherel-Rotach-type asymptotics for the orthogonal polynomials found in [ 12] plays a central role. The formulae in [46] express the correlation kernels for beta = 1, 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 [9], 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 [9]. (c) 2006 Wiley Periodicals, Inc.