Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge
type: Item
instance of: scholarly article
title: Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge
zbMATH Open document ID: 1371.33034
author: A. Bogatskiy
author: Tom Claeys
author: A. R. Its
published in: Communications in Mathematical Physics
publication date: 2016-10-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1507.01710
review text: 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…
reviewed by: Edmundo Huertas
Mathematics Subject Classification ID: 33E17
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 60B20
zbMATH DE Number: 6641311
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: Painlevé transcendents
zbMATH Keywords: Riemann-Hilbert problem
zbMATH Keywords: random matrix theory
MaRDI profile type: Publication
OpenAlex ID: W845931339
arXiv ID: 1507.01710
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DOI: 10.1007/S00220-016-2691-Y
description: scientific article; zbMATH DE number 6641311