Quadratic Vector Equations on Complex Upper Half-Plane
Dec 2019 · American Mathematical Soc.
Ebook
133
Pages
About this ebook
The authors consider the nonlinear equation −1m=z+Sm with a parameter z in the complex upper half plane H, where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on 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∈H, including the vicinity of the singularities.
About the author
Oskari Ajanki: Institute of Science and Technology, Klosterneuberg, Austria,
László Erdős: Institute of Science and Technology, Klosterneuberg, Austria
Torben Krüger: Institute of Science and Technology, Klosterneuberg, Austria
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