Skew-orthogonal polynomials, differential systems and random matrix theory

Abstract

We study skew-orthogonal polynomials with respect to the weight function $\exp [-2V(x)]$, with $V(x)={\sum }_{K=1}^{2d}(u_K/K)x^K$, $u_{2d}>0$, $d>0$. A finite subsequence of such skew-orthogonal polynomials arising in the study of Orthogonal and Symplectic ensembles of random matrices, satisfy a system of differential-difference-deformation equation. The vectors formed by such subsequence has the rank equal to the degree of the potential in the quaternion sense. These solutions satisfy certain compatibility condition and hence admit a simultaneous fundamental system of solutions.

30 pages

References in corpus (6)

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• Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles
• Mixed correlation function and spectral curve for the 2-matrix model
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Cited by in corpus (4)

• Generalized Christoffel-Darboux formula for classical skew-orthogonal polynomials
• A Riemann-Hilbert problem for skew-orthogonal polynomials
• Skew-orthogonal polynomials: the quartic case
• Bulk asymptotics of skew-orthogonal polynomials for quartic double well potential and universality in the matrix model