2006/05/08 by Maurice Duits, M Duits, Arno Kuijlaars +1 · 2 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Mathematical functions and polynomials #Nonlinear Waves and Solitons #math-ph #math.CA #math.MP
paper · pdf · doi:10.1088/0951-7715/19/10/001
published as Nonlinearity 19 (2006), 2211--2245 · 52 pages, 10 figures
arxiv created 2006/05/08 · openalex publication_date 2006/08/18 · arxiv updated 2010/07/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
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 Painlevé 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 Ψ-functions for the Painlevé I equation in the construction of the local parametrix.