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.

1 aBertola, Marco1 aElias Rebelo, José Gustavo1 aGrava, Tamara uhttps://www.math.sissa.it/publication/painlev%C3%A9-iv-critical-asymptotics-orthogonal-polynomials-complex-plane