01288nas a2200145   4500008004100000245008600041210007000127260004400197490000700241520070800248100001900956700003200975700001801007856011701025       2018                        eng d00aPainlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane0 aPainlevé IV Critical Asymptotics for Orthogonal Polynomials in t  bNational Academy of Sciences of Ukraine0 v143 a<p>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.</p>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