Painlev\'e I asymptotics for orthogonal polynomials with respect to a varying quartic weight
Seminar Room 1, Newton Institute
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 Painlev´e 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 equilibrium problem and the use of $\Psi$-functions for the Painlev´e I equation in the construction of the local parametrix.