TY - JOUR

T1 - Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices

AU - Deift, Percy

AU - Gioev, Dmitri

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2007/6

Y1 - 2007/6

N2 - We prove universality at the edge of the spectrum for unitary (β = 2), orthogonal (β = 1), and symplectic (β = 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) = κ2mx2m + ⋯, κ2m > 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 [12], 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 β= 1, 4 as a sum of a Christoffel-Darboux (CD) term, as in the case β = 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].

AB - We prove universality at the edge of the spectrum for unitary (β = 2), orthogonal (β = 1), and symplectic (β = 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) = κ2mx2m + ⋯, κ2m > 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 [12], 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 β= 1, 4 as a sum of a Christoffel-Darboux (CD) term, as in the case β = 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].

UR - http://www.scopus.com/inward/record.url?scp=34247509776&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34247509776&partnerID=8YFLogxK

U2 - 10.1002/cpa.20164

DO - 10.1002/cpa.20164

M3 - Article

AN - SCOPUS:34247509776

VL - 60

SP - 867

EP - 910

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 6

ER -