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Quadratic Vector Equations on Complex Upper Half-Plane
Author(s): Laszlo Erdos (author) & Torben Kruger (author) Oskari Ajanki (author) (Author)
- Publisher: American Mathematical Society
- Publication Date: 30 Oct. 2019
- Language: English
- Print length: 132 pages
- ISBN-10: 1470436833
- ISBN-13: 9781470436834
Book Description
The authors consider the nonlinear equation $-\frac 1m=z+Sm$ 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 a previous paper the authors 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 the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They 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 authors' companion paper they present a complete stability analysis of the equation for any $z\in \mathbb H$, including the vicinity of the singularities.
Editorial Reviews
About the Author
Oskari Ajanki, Institute of Science and Technology, Klosterneuberg, Austria.
Laszlo Erdos, Institute of Science and Technology, Klosterneuberg, Austria.
Torben Kruger, Institute of Science and Technology, Klosterneuberg, Austria.
