Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

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.

References in corpus (4)

• Universality of a double scaling limit near singular edge points in random matrix models
• Shocks and finite-time singularities in Hele-Shaw flow
• Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices
• Random Normal Matrices and Polynomial Curves

Cited by in corpus (10)

• The random normal matrix model: insertion of a point charge
• Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges
• Exponential moments for disk counting statistics of random normal matrices in the critical regime
• Scaling Limits of Planar Symplectic Ensembles
• Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials
• Lemniscate ensembles with spectral singularity
• Fredholm determinant representation of the Painlevé II $\tau$-function
• The $\tau$-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant
• Orthogonal polynomials in the normal matrix model with two insertions
• Szegő type asymptotics for the reproducing kernel in spaces of full-plane weighted polynomials