Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

Symmetry Integrability and Geometry-Methods and Applications

article

Marco Bertola¹  José Gustavo Elias Rebelo²  Tamara Grava² 

[1] SISSA;Department of Mathematics and Statistics, Concordia University

关键词: orthogonal polynomials on the complex plane;    Riemann–Hilbert problem;    Painlev´e equations Fredholm determinant;   

DOI  :  10.3842/SIGMA.2018.091

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown