The Schwarz function and the shrinking of the Szeg\H{o} curve: electrostatic, hydrodynamic, and random matrix models

TL;DR

This paper explores the deformation of the Szeg\

Contribution

It introduces a unified framework connecting electrostatic, hydrodynamic, and random matrix models via the asymptotic zero distribution of Laguerre polynomials, with explicit Schwarz function expressions.

Findings

Schwarz functions of the curves are expressed using Lambert W function.

The deformation parameter t encodes the approximation rate of alpha_n to negative integers.

The S-property is explicitly described as Schwarz reflection symmetry.

Abstract

We study the deformation of the classical Szeg\H{o} curve $\gamma_0$ given by $\gamma_t = \{ z\in\mathbb{C}: |z\, e^{1-z}| = e^{-t}, |z|\leq 1\}$, $t\geq 0$ from three different viewpoints: an electrostatic equilibrium problem, the dual hydrodynamic model, and a random matrix model. The common framework underlying these models is the asymptotic distribution of zeros of the scaled varying Laguerre polynomials $L^{(\alpha_n)}_n(n z)$ in the critical regime where $\lim_{n\to\infty}\alpha_n/n=-1$, for which the limiting zero distribution is supported on $\gamma_t$, where the deformation parameter $t$ encodes the exponential rate at which the sequence $\alpha_n$ approximates the set of negative integers. We show that the Schwarz functions of these curves can be written in terms of the Lambert $W$ function, and that in this formulation the $S$-property of Stahl and Gonchar and Rachmanov can be explictly written as the Schwarz reflection symmetry. We also discuss a conformal map of the interior of the curves $\gamma_t$ onto the disks $D(0,e^{-t})$ and the harmonic moments of the curves.