The Schwarz function and the shrinking of the Szegő curve: electrostatic, hydrodynamic, and random matrix models
Year: 2026
Type: article
Abstract: We study the deformation of the classical Szegő curve $$\gamma _0$$ γ 0 given by $$\gamma _t = \{ z\in \mathbb {C}: |z\, e^{1-z}| = e^{-t}, |z|\le 1\}$$ γ t = { z ∈ C : | z e 1 - z | = e - t , | z | ≤ 1 } , $$t\ge 0$$ 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 $$L^{(\alpha _n)}_n(n z)$$ L n ( α n ) ( n z ) in the critical regime where $$\lim _{n\rightarrow \infty }\alpha _n/n=-1$$ lim n → ∞ α n / n = - 1 , for which the limiting zero distribution is supported on $$\gamma _t$$ γ t , where the deformation parameter t encodes the exponential rate at which the sequence $$\alpha _n$$ 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 $$\gamma _t$$ γ t onto the disks $$D(0,e^{-t})$$ D ( 0 , e - t ) and the harmonic moments of the curves.
Institutions: Universidad Complutense de Madrid, Universidad de Cádiz
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Awards: G1429215307, G2580175713, G717256432
OpenAlex ID: W7150959886