### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 091, 34 pages      arXiv:1802.01153      https://doi.org/10.3842/SIGMA.2018.091
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

### Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

Marco Bertola ab, José Gustavo Elias Rebelo a and Tamara Grava ac
a) Area of Mathematics, SISSA, via Bonomea 265 - 34136, Trieste, Italy
b) Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, Québec, Canada H3G 1M8
c) School of Mathematics, University of Bristol, UK

Received February 06, 2018, in final form August 14, 2018; Published online August 30, 2018

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.

Key words: orthogonal polynomials on the complex plane; Riemann-Hilbert problem; Painlevé equations Fredholm determinant.

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