Hankel determinants for a singular complex weight and the first and third Painlevé transcendents

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)64-92
Journal / PublicationJournal of Approximation Theory
Online published3 Feb 2016
Publication statusPublished - May 2016


In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight en(z−log z+t/z) for t < 0 and z on certain contours in the complex plane. When the parameters n, t and the degree k are fixed, the Hankel determinant for the singular complex weight is shown to be the isomonodromy τ-function of the Painlevé III equation. When the degree k = n, n is large and t is close to a critical value, inspired by the study of the Wigner time delay in quantum transport, we show that the double scaling asymptotic behaviors of the recurrence coefficients and the Hankel determinant are described in terms of a Boutroux tronquée solution to the Painlevé I equation. Our approach is based on the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.

Research Area(s)

  • Asymptotics, Hankel determinants, Painlevé I equation, Painlevé III equation, Riemann-Hilbert approach