Quadratic Vector Equations on Complex Upper Half-Plane (Memoirs of the American Mathematical Society) - Softcover

Oskari Ajanki (author), Laszlo Erdos (author) & Torben Kruger (author)

Softcover

ISBN 10: 1470436833 ISBN 13: 9781470436834

Publisher: American Mathematical Society, 2019

Synopsis

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.

About this title

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.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2019
• Language: English
• ISBN 10: 1470436833
• ISBN 13: 9781470436834
• Binding: Paperback
• Number of pages: 132