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.
|Number of pages||34|
|Journal||Symmetry, Integrability and Geometry: Methods and Applications|
|Early online date||30 Aug 2018|
|Publication status||Published - 2018|
- orthogonal polynomials on the complex plane
- Riemann-Hilbert problems
- Strong asymptotics