Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge
After the pioneer works of \textit{A. S. Fokas} et al. [Commun. Math. Phys. 142, No. 2, 313--344 (1991; Zbl 0742.35047); ibid. 147, No. 2, 395--430 (1992; Zbl 0760.35051)], and \textit{A. P. Magnus} (continuous case) [J. Comput. Appl. Math. 57, No. 1--2, 215--237 (1995; Zbl 0828.42012)], it is very well known that there exists a close relationship between orthogonal polynomials and Painlevé equations. Painlevé equations can be seen as analogues of the classical special functions for \textit{nonlinear} ordinary differential equations. They were first discovered by Paul Painlevé (1863--1933) and his collaborators around 1900. They possess the interesting property that their solutions have no movable branch points, and the only movable singularities are poles, which is known as the \textit{Painlevé property}. Painlevé proved that, under certain transformations, every second order differential equation of the form \[ \frac{d^{2}y}{dz^{2}}=F\left( \frac{dy}{dz},y,z\right) \] (where \(F\) is a rational function) can be put into one of fifty canonical forms. Forty-four of them are solvable in terms of elementary known functions (\textit{transcendents}), and the remaining six equations (labeled from \(\mathrm{P}_{I}\) to \(\mathrm{P}_{VI}\)) require new special functions to solve them, which are called the \textit{six Painlevé transcendents}. This beautiful paper connects deep interesting and intricate mathematical results with problems in mathematical physics and random matrix theory. In the first part of the paper, the authors describe the large \(n\) asymptotic behavior, in the transition regime where the point of discontinuity \(\lambda _{0}(n)=\lambda \sqrt{2n}\), with \(\lambda =1+\frac{t}{2}n^{-2/3}\), \(t\in \mathbb{R}\), for the sequence of monic polynomials orthogonal with respect to the discontinuous Gaussian weight \[ w(x)=e^{-x^{2}}\times \begin{cases} e^{\pi i\beta }, & x<\lambda _{0}(n) \\ e^{-\pi i\beta } & x\geq \lambda _{0}(n) \end{cases},\quad \lambda _{0}(n)\in \mathbb{R}, \] their corresponding coefficients of their three term recurrence relation, and their associated Hankel determinants. They restrict themselves to the case Re\(\,\beta \in \left( -\frac{1}{2},\frac{1}{2}\right] \), because that weight function is periodic in \(\beta \), and it is positive in case \(\beta\) is a pure imaginary number. They describe this behavior in terms of the Ablowitz-Segur solutions to the \(\mathrm{P}_{II}\) equation, and they are then able to find a new result on the poles of the Ablowitz-Segur solutions to the \(\mathrm{P}_{II}\) equation and to conjecture large \(n\) asymptotics for an Airy kernel Fredholm determinant and total integral identities for \textit{Painlevé II transcendents}. It is worth noting that their results complement the ones in [\textit{S.-X. Xu} and \textit{Y.-Q. Zhao}, Stud. Appl. Math. 127, No. 1, 67--105 (2011; Zbl 1225.33030)], where the authors study the influence of the discontinuities on the asymptotic behavior of recurrence coefficients for infinite sequences of orthogonal polynomials, and based on a result in this paper, they prove a new result about the poles for those \(\mathrm{P}_{II}\) transcendents. Finally, they also show some nice applications of their results in random matrix theory.
scientific article; zbMATH DE number 6641311
scientific article