Asymptotics for a class of planar orthogonal polynomials and truncated unitary matrices

TL;DR

This paper analyzes the asymptotic behavior of a class of planar orthogonal polynomials related to non-Hermitian random matrices, using Riemann-Hilbert techniques to derive detailed asymptotics and partition function expansions.

Contribution

It provides the first comprehensive asymptotic analysis of these orthogonal polynomials with a complex measure, extending previous Gaussian weight results.

Findings

Asymptotics obtained in all regions of the complex plane

Partition function asymptotic expansion derived

Method converts planar orthogonality to contour-based analysis

Abstract

We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure \begin{equation*} d\mu(z) = (1-|z|^{2})^{\alpha-1}|z-x|^{\gamma}\mathbf{1}_{|z| < 1}d^{2}z, \end{equation*} where $d^{2}z$ is the two dimensional area measure, $\alpha$ is a parameter that can grow with $n$, while $\gamma>-2$ and $x>0$ are fixed. This measure arises naturally in the study of characteristic polynomials of non-Hermitian ensembles and generalises the example of a Gaussian weight that was recently studied by several authors. We obtain asymptotics in all regions of the complex plane and via an appropriate differential identity, we obtain the asymptotic expansion of the partition function. The main approach is to convert the planar orthogonality to one defined on suitable contours in the complex plane. Then the asymptotic analysis is performed using the Deift-Zhou steepest descent method for the associated Riemann-Hilbert problem.