Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters

TL;DR

This paper investigates the large-degree asymptotics of Laguerre polynomials with complex parameters, utilizing quadratic differentials and Riemann-Hilbert techniques to handle the added complexity of complex asymptotic behavior.

Contribution

It extends asymptotic analysis of Laguerre polynomials to complex parameters, introducing new contour analysis via quadratic differentials and advanced Riemann-Hilbert methods.

Findings

Derived weak asymptotics using Gonchar-Rakhmanov-Stahl theorem.

Established strong asymptotics through non-commutative steepest descent.

Analyzed the global structure of quadratic differential trajectories.

Abstract

In this paper we study the asymptotics (as $n\to \infty$) of the sequences of Laguerre polynomials with varying complex parameters $\alpha$ depending on the degree $n$. More precisely, we assume that $\alpha_n = n A_n, $ and $ \lim_n A_n=A \in \mathbb{C}$. This study has been carried out previously only for $\alpha_n\in \mathbb{R}$, but complex values of $A$ introduce an asymmetry that makes the problem more difficult. The main ingredient of the asymptotic analysis is the right choice of the contour of orthogonality, which requires the analysis of the global structure of trajectories of an associated quadratic differential on the complex plane, which may have an independent interest. While the weak asymptotics is obtained by reduction to the theorem of Gonchar--Rakhmanov--Stahl, the strong asymptotic results are derived via the non-commutative steepest descent analysis based on the Riemann-Hilbert characterization of the Laguerre polynomials.