Painlevé IV equation, Fredholm Determinant and Double-Scaling Limits

This thesis focuses on the Painlevé IV equation and its relationship with the to double scaling limits in a normal matrix models in a potential with a discrete rotational symmetry. In the first part we study a special solution of the Painlevé IV equation, which is determined by a particular choice of the monodromy data of the associated linear system, and consider the Riemann-Hilbert Problem associated to it. From the Riemann-Hilbert problem we use the theory of integrable operators in order to associate a Fredholm determinant representation, or equivalently a tau-function, to our specific solution. The poles of the Painlevé solution are the zeros of this tau-function. We study numerically the tau-function for real values of the independent variable s and locate its zero on the real line. In the second part of the thesis we introduce the subject of orthogonal polynomials that appear in the study of statistical quantities related to normal matrix models. We chose a potential with a discrete rotational symmetry. A potential of this form has different regimes: pre-critical, critical and post-critical, according to the values of its parameter. Such regimes describe the transition of the support of the limiting distribution of the eigenvalues of the normal matrix model from a connected domain to a domain with several connected components. We are interested in considering the critical case. Our purpose is to consider the orthogonal polynomials associated with this Matrix Model and study their asymptotic behaviour. We achieve this goal by transforming the orthogonality relations on the complex plane to a Riemann-Hilbert Problem on a contour. By following the general method of nonlinear steepest descent of Deift-Zhou we are able to perform the asymptotic analysis of the orthogonal polynomials as the degree of the polynomials goes to infinity. As a consequence of this procedure, we will find that the Riemann-Hilbert Problem obtained after some transformations is the same as the one that we had obtained for our special solution to Painlevé IV equation. We will then fulfill our goal of finding the asymptotic behaviour of the orthogonal polynomials in every region of the complex plane.