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.To the memory of Andrei Kapaev, a master of Painlevé equation, a very generous colleague and a special friend.

Scaling limits of random normal matrix processes at singular boundary points

“…Ordinary boundary points have been classified by Sakai [23], providing a suitable platform to study such points in complete generality. At this point, there does not seem to exist a similar classification of special boundary points, and we shall merely compare with some examples of such points, which emerge naturally in the recent papers [7,14].…”

“…It is easy to see [7,14] that the droplet S corresponding to Q is the interior of the lemniscate |ζ k − 1/ √ k| = 1/ √ k, while the equilibrium measure is given by the density k 2 |ζ| 2k−2 1 S (ζ). In particular, 0 ∈ ∂S and ∆Q(0) = 0, so the origin is a special singular boundary point.…”

“…Figure 2. Another type of singularity, a crossing point, may emerge at a boundary point where ∆Q = 0, as in the example of the lemniscate ensemble [7]. In this case we zoom at the singular point itself, but due to the vanishing of the equilibrium density, we require a relatively coarse scale in order to recover a nontrivial scaling limit.…”

Characteristic Polynomials of Complex Random Matrices and Painlevé Transcendents

“…In the recent work [18] the orthogonal polynomials for the planar weight e −N V (d) (λ,t) have been analysed. Close to the transition t = t c the asymptotics were expressed in terms of the Hamiltonian of the Painlevé IV system.…”

The τ-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant

“…This approach cannot be directly implemented to the cases where the RHP is formulated on a contour that is not a circle as is the case for the Painlevé equations PI, PII and PIV. There are several examples of τ -functions expressed as a Fredholm determinant like the Tracy-Widom distribution related to Painlevé II [22], or like the example obtained in [3] related to Painlevé IV. However the generic τ -function of the Painlevé I, II and IV equations does not seem to have a Fredholm determinant representation.…”