A Riemann-Hilbert problem for skew-orthogonal polynomials

Details A Riemann-Hilbert problem for skew-orthogonal polynomials A Riemann-Hilbert problem for skew-orthogonal polynomials

2006-10-19

arxiv.org/abs/math/0610592v3

Mathematics

Complex Variables

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

Scientific paper

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.

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