Abstract

We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to a certain normal matrix model. The model depends on a parameter and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlevé IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real line, showing also that the corresponding Painlevé transcendent is pole-free on a semiaxis.

Details

Title
Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane
Author
Bertola, Marco; José Gustavo Elias Rebelo; Grava, Tamara
Publication year
2018
Publication date
2018
Publisher
National Academy of Sciences of Ukraine
e-ISSN
18150659
Source type
Scholarly Journal
Language of publication
English
ProQuest document ID
2097998669
Copyright
© 2018. This work is published under http://creativecommons.org/licenses/by-sa/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.