A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type

TL;DR

This paper develops a Riemann-Hilbert framework for skew-orthogonal polynomials of symplectic type, enabling new formulas and integrable system analogues for $eta=4$ random matrix ensembles.

Contribution

It introduces a Riemann-Hilbert problem representation for skew-orthogonal polynomials of symplectic type, extending the Christoffel-Darboux formula and deriving a Toda lattice analogue.

Findings

Riemann-Hilbert representation for skew-orthogonal polynomials

Beta=4 Christoffel-Darboux formula

Lax pair leading to Toda lattice analogue

Abstract

We present a representation of skew-orthogonal polynomials of symplectic type ($\beta=4$) in terms of a matrix Riemann-Hilbert problem, for weights of the form ${\rm e}^{-V(z)}$ where $V$ is a polynomial of even degree and positive leading coefficient. This is done by representing skew-orthogonality as a kind of multiple-orthogonality. From this, we derive a $\beta=4$ analogue of the Christoffel-Darboux formula. Finally, our Riemann-Hilbert representation allows us to derive a Lax pair whose compatibility condition may be viewed as a $\beta=4$ analogue of the Toda lattice.