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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.07832 |
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| _version_ | 1866914461173940224 |
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| author | Álvarez, Gabriel Alonso, Luis Martínez Medina, Elena |
| author_facet | Álvarez, Gabriel Alonso, Luis Martínez Medina, Elena |
| contents | We study the deformation of the classical Szegő curve $γ_0$ given by $γ_t = \{ z\in\mathbb{C}: |z\, e^{1-z}| = e^{-t}, |z|\leq 1\}$, $t\geq 0$ 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 $L^{(α_n)}_n(n z)$ in the critical regime where $\lim_{n\to\infty}α_n/n=-1$, for which the limiting zero distribution is supported on $γ_t$, where the deformation parameter $t$ encodes the exponential rate at which the sequence $α_n$ 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 $γ_t$ onto the disks $D(0,e^{-t})$ and the harmonic moments of the curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07832 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Schwarz function and the shrinking of the Szegő curve: electrostatic, hydrodynamic, and random matrix models Álvarez, Gabriel Alonso, Luis Martínez Medina, Elena Mathematical Physics We study the deformation of the classical Szegő curve $γ_0$ given by $γ_t = \{ z\in\mathbb{C}: |z\, e^{1-z}| = e^{-t}, |z|\leq 1\}$, $t\geq 0$ 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 $L^{(α_n)}_n(n z)$ in the critical regime where $\lim_{n\to\infty}α_n/n=-1$, for which the limiting zero distribution is supported on $γ_t$, where the deformation parameter $t$ encodes the exponential rate at which the sequence $α_n$ 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 $γ_t$ onto the disks $D(0,e^{-t})$ and the harmonic moments of the curves. |
| title | The Schwarz function and the shrinking of the Szegő curve: electrostatic, hydrodynamic, and random matrix models |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2604.07832 |