Painleve I asymptotics for orthogonal polynomials with respect to a varying quartic weight
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.
Keywords
Cite
@article{arxiv.math/0605201,
title = {Painleve I asymptotics for orthogonal polynomials with respect to a varying quartic weight},
author = {Maurice Duits and Arno Kuijlaars},
journal= {arXiv preprint arXiv:math/0605201},
year = {2010}
}
Comments
52 pages, 10 figures