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

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.