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

TL;DR

This paper proves the universality of spectral edge behavior for various classes of random matrix ensembles with polynomial weights, extending known results to orthogonal and symplectic cases.

Contribution

It establishes edge universality for orthogonal and symplectic ensembles with polynomial weights, building on prior work for unitary ensembles and addressing technical challenges at the spectral edge.

Findings

Universality at the spectral edge for beta=1, 2, 4 ensembles.

Extension of edge universality results to orthogonal and symplectic cases.

Technical development handling correction terms in the asymptotics.

Abstract

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].