Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

Marco Bertola, Jose Gustavo Elias Rebelo, Tamara Grava

Research output: Contribution to journalArticle (Academic Journal)peer-review

6 Citations (Scopus)
193 Downloads (Pure)

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.
Original languageEnglish
Article number091
Number of pages34
JournalSymmetry, Integrability and Geometry: Methods and Applications
Volume14
Early online date30 Aug 2018
DOIs
Publication statusPublished - 2018

Keywords

  • orthogonal polynomials on the complex plane
  • Riemann-Hilbert problems
  • Strong asymptotics

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