Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2018
• Main Authors: Bertola, M., Rebelo, J.G.E., Grava, T.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2018
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/209866
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane / M. Bertola, J.G.E. Rebelo, T. Grava // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 43 назв. — англ.

We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated with 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 with such a solution, and we compute it numerically on the real line, showing also that the corresponding Painlevé transcendent is pole-free on a semiaxis.