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

Abstract We study the deformation of the classical Szegő curve $$\gamma _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> given by $$\gamma _t = \{ z\in \mathbb {C}: |z\, e^{1-z}| = e^{-t}, |z|\le 1\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>z</mml:mi> <mml:mo>∈</mm