Orthonormal Polynomials on the Unit Circle and Spatially Discrete Painlevé II Equation

Abstract

We consider the polynomials ${\varphi }_n(z)={\kappa }_n(z^n+b_{n-1}{z}^{n-1}+>...)$ orthonormal with respect to the weight $\exp (\sqrt{\lambda }(z+1/z))dz/2\pi iz$ on the unit circle in the complex plane. The leading coefficient ${\kappa }_n$ is found to satisfy a difference-differential (spatially discrete) equation which is further proved to approach a third order differential equation by double scaling. The third order differential equation is equivalent to the Painlevé II equation. The leading coefficient and second leading coefficient of ${\varphi }_n(z)$ can be expressed asymptotically in terms of the Painlevé II function.

16 pages

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• Level-Spacing Distributions and the Airy Kernel
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Cited by in corpus (1)

• Riemann-Hilbert problems for last passage percolation