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A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type
Alex Little Unité de Mathématiques Pures et Appliquées, ENS de Lyon, France
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.
Keywords:
Riemann–Hilbert problem, skew-orthogonal polynomials, random matrices.
Received: December 27, 2023; in final form August 6, 2024; Published online August 16, 2024
Citation:
Alex Little, “A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type”, SIGMA, 20 (2024), 076, 32 pp.
Linking options:
https://www.mathnet.ru/eng/sigma2078 https://www.mathnet.ru/eng/sigma/v20/p76
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