Hankel determinants for a singular complex weight and the first and third Painlev\'e transcendents

TL;DR

This paper investigates Hankel determinants associated with a singular complex weight, revealing their connection to Painlevé equations and analyzing their asymptotic behavior using Riemann-Hilbert techniques.

Contribution

It establishes a link between Hankel determinants for a perturbed Laguerre weight and Painlevé III and I equations, providing asymptotic analysis near critical points.

Findings

Hankel determinant equals the isomonodromy τ-function of Painlevé III.

Double scaling limits described by a Boutroux tronquée solution to Painlevé I.

Asymptotic behavior analyzed using Deift-Zhou steepest descent method.

Abstract

In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight $e^{-n(z-\log z+t/z)}$ 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 $\tau$-function of the Painlev\'e 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 tronqu\'ee solution to the Painlev\'e I equation. Our approach is based on the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems.