The Schwarz function and the shrinking of the Szegő curve: electrostatic, hydrodynamic, and random matrix models

We study the deformation of the classical Szegő curve γ ₀ γ 0 given by γ ₜ = \ z∈ C: |z\, e¹⁻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(α ₙ)ₙ(n z) L n ( α n ) ( n z ) in the critical regime where lim n→ ∞α ₙ/n=-1 lim n → ∞ α n / n = - 1 , for which the limiting zero distribution is supported on γ ₜ γ t , where the deformation parameter t encodes the exponential rate at which the sequence α 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 γ ₜ γ t onto the disks D(0,e⁻ᵗ) D ( 0 , e - t ) and the harmonic moments of the curves.