TY - JOUR T1 - Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane JF - Symmetry, Integrability and Geometry. Methods and Applications Y1 - 2018 A1 - Marco Bertola A1 - José Gustavo Elias Rebelo A1 - Tamara Grava AB -

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 Painlev´e 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 Painlev´e transcendent is pole-free on a semiaxis.

PB - National Academy of Sciences of Ukraine VL - 14 ER -