A Riemann-Hilbert problem for skew-orthogonal polynomials

Abstract

We find a local $(d+1)\times (d+1)$ Riemann-Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of the Gaussian Orthogonal Ensemble of random matrices with a potential function of degree $d$. Our Riemann-Hilbert problem is similar to a local $d\times d$ Riemann-Hilbert problem found by Kuijlaars and McLaughlin characterizing the bi-orthogonal polynomials. This gives more motivation for finding methods to compute asymptotics of high order Riemann-Hilbert problems, and brings us closer to finding asymptotics of the skew-orthogonal polynomials.

9 pages, v. 2 fixed some typos, v. 3 revised proof

References in corpus (6)

• Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices
• Classical skew orthogonal polynomials and random matrices
• Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem
• Universality in Random Matrix Theory for orthogonal and symplectic ensembles
• Skew-orthogonal polynomials, differential systems and random matrix theory
• Biorthogonal polynomials for 2-matrix models with semiclassical potentials

Cited by in corpus (1)

• Skew-orthogonal polynomials, differential systems and random matrix theory