Strong asymptotics of the orthogonal polynomials with respect to a measure supported on the plane

TL;DR

This paper derives strong asymptotics and zero distribution of orthogonal polynomials on the complex plane with a specific measure, revealing phase transitions and connections to Painlevé II in a double scaling limit.

Contribution

It provides the first detailed asymptotic analysis of complex-plane orthogonal polynomials with a non-Hermitian measure, including phase transition behavior and Painlevé II connection.

Findings

Asymptotics described by three associated measures.

Identification of a topological transition in the support region.

Appearance of Painlevé II solution near critical parameter.

Abstract

We consider the orthogonal polynomials $\{P_{n}(z)\}$ with respect to the measure $|z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$ over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where $n$ grows to infinity with $N$. The asymptotics are described in terms of three (probability) measures associated with the problem. The first measure is the limit of the counting measure of zeros of the polynomials, which is captured by the $g$-function much in the spirit of ordinary orthogonal polynomials on the real line. The second measure is the equilibrium measure that minimizes a certain logarithmic potential energy, supported on a region $K$ of the complex plane. The third measure is the harmonic measure of $K^c$ with a pole at $\infty$. This appears as the limit of the probability measure given (up to the normalization constant) by the squared modulus of the $n$-th orthogonal polynomial times the orthogonality measure, i.e. $|P_n(z)|^2 |z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$. The compact region $K$ that is the support of the second measure undergoes a topological transition under the variation of the parameter $t=n/N$; in a double scaling limit near the critical point given by $t_c=a(a+2\sqrt c)$ we observe the Hastings-McLeod solution to Painlev\'e\ II in the asymptotics of the orthogonal polynomials.