Physics – Mathematical Physics
Scientific paper
2005-07-08
Comm. Pure Appl. Math. 60 (2007), no. 6, 867-910
Physics
Mathematical Physics
36 pages
Scientific paper
10.1002/cpa.20164
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].
Deift Percy
Gioev Dimitri
No associations
LandOfFree
Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-375112