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 (2)

• Universality in Random Matrix Theory for orthogonal and symplectic ensembles
• Biorthogonal polynomials for 2-matrix models with semiclassical potentials