We study the deformation of the classical Szegő curve <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\gamma _0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> given by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma _t = \{ z\in \mathbb {C}: |z\, e^{1-z}| = e^{-t}, |z|\le 1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mi>t</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mo stretchy="false">|</mo> <mi>z</mi> <mspace width="0.166667em" /> </mrow> <msup> <mi>e</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>z</mi> </mrow> </msup> <mrow> <mo stretchy="false">|</mo> <mo>=</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>t</mi> </mrow> </msup> <mrow> <mo>,</mo> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> <mo>≤</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$t\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^{(\alpha _n)}_n(n z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in the critical regime where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lim _{n\rightarrow \infty }\alpha _n/n=-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo movablelimits="true">lim</mo> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </msub> <msub> <mi>α</mi> <mi>n</mi> </msub> <mo stretchy="false">/</mo> <mi>n</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, for which the limiting zero distribution is supported on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma _t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation>, where the deformation parameter t encodes the exponential rate at which the sequence <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha _n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>α</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\gamma _t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> onto the disks <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$D(0,e^{-t})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the harmonic moments of the curves.

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

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2026年4月7日

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Gabriel Álvarez,
Luis Martínez Alonso,
Elena Medina

摘要

We study the deformation of the classical Szegő curve $\gamma _0$ $γ_{0}$ given by $\gamma _t = \{ z\in \mathbb {C}: |z\, e^{1-z}| = e^{-t}, |z|\le 1\}$ $γ_{t} = {z \in C : | z e^{1 - z} | = e^{- t}, | z | \leq 1}$ , $t\ge 0$ $t \geq 0$ 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 $L^{(\alpha _n)}_n(n z)$ $L_{n}^{(α_{n})} (n z)$ in the critical regime where $\lim _{n\rightarrow \infty }\alpha _n/n=-1$ ${lim}_{n \to \infty} α_{n} / n = - 1$ , 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.