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Painleve I asymptotics for orthogonal polynomials with respect to a varying quartic weight

Maurice Duits·Arno Kuijlaars
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Abstract

We study polynomials that are orthogonal with respect to a varying quartic weight \exp(-N(x^2/2+tx^4/4)) for t<0, where the orthogonality takes place on certain contours in the complex plane. Inspired by developments in 2D quantum gravity, Fokas, Its, and Kitaev, showed that there exists a critical value for t around which the asymptotics of the recurrence coefficients are described in terms of exactly specified solutions of the Painleve I equation. In this paper, we present an alternative and more direct proof of this result by means of the Deift/Zhou steepest descent analysis of the Riemann-Hilbert problem associated with the polynomials. Moreover, we extend the analysis to non-symmetric combinations of contours. Special features in the steepest descent analysis are a modified equililbrium problem and the use of Psi-functions for the Painleve I equation in the construction of the local parametrix.

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Maurice Duits, Arno Kuijlaars. 2006-05-08. Painleve I asymptotics for orthogonal polynomials with respect to a varying quartic weight. https://doi.org/10.1088/0951-7715%2F19%2F10%2F001

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