Title: Painleve I asymptotics for orthogonal polynomials with respect to a varying quartic weight
Authors: Duits, Maurice ×
Kuijlaars, Arno #
Issue Date: 2006
Publisher: Iop publishing ltd
Series Title: Nonlinearity vol:19 issue:10 pages:2211-2245
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.
ISSN: 0951-7715
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Analysis Section
× corresponding author
# (joint) last author

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