Abstract We study the deformation of the classical Szegő curve ₀ γ 0 given by ₜ = \ z {C: |z\, e^1-z| = e^-t, |z| 1\} γ t = z ∈ C: | z e 1 - z | = e - t, | z | ≤ 1, t 0 t ≥ 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^ (ₙ) ₙ (n z) L n (α n) (n z) in the critical regime where ₍ ₙ/n=-1 lim n → ∞ α n / n = - 1, for which the limiting zero distribution is supported on ₜ γ t, where the deformation parameter t encodes the exponential rate at which the sequence ₙ α 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 ₜ γ t onto the disks D (0, e^-t) D (0, e - t) and the harmonic moments of the curves.
Álvarez et al. (Tue,) studied this question.