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arXiv · 1506.05095

Quadratic vector equations on complex upper half-plane

Oskari Ajanki·Laszlo Erdos·Torben Krüger
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Abstract

We consider the nonlinear equation $-\frac{1}{m}=z+Sm$ with a parameter $z$ in the complex upper half plane $\mathbb{H} $, where $S$ is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in $ \mathbb{H}$ is unique and its $z$-dependence is conveniently described as the Stieltjes transforms of a family of measures $v$ on $\mathbb{R}$. In [AEK17a] we qualitatively identified the possible singular behaviors of $v$: under suitable conditions on $S$ we showed that in the density of $v$ only algebraic singularities of degree two or three may occur. In this paper we give a comprehensive analysis of these singularities with uniform quantitative controls. We also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the companion paper [AEK16b], we present a complete stability analysis of the equation for any $z\in \mathbb{H}$, including the vicinity of the singularities.

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Oskari Ajanki, Laszlo Erdos, Torben Krüger. 2020-06-09. Quadratic vector equations on complex upper half-plane. https://doi.org/10.1090/memo%2F1261

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