2005/07/09 by Percy Deift, Dimitri Gioev · 8 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Random Matrices and Applications #math-ph #math.CA #math.MP #msc:15A52
paper · pdf · doi:10.1002/cpa.20164
published as Comm. Pure Appl. Math. 60 (2007), no. 6, 867-910 · 36 pages
arxiv created 2005/07/09 · openalex publication_date 2006/10/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Abstract 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 ) = κ 2 m x 2 m + · · ·, κ 2 m > 0. The precise statement of our results is given in Theorem 1.1 and Corollaries 1.2 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 [49]. © 2006 Wiley Periodicals, Inc.