Quadratic Vector Equations on Complex Upper Half-Plane (Memoirs of the American Mathematical Society, September 2019, Number 1261) - Softcover

Ajanki, Oskari Heikki; Erdos, Laszlo; Kruger, Torben

Softcover

ISBN 10: 1470436833 ISBN 13: 9781470436834

Publisher: Amer Mathematical Society

Synopsis

The authors consider the nonlinear equation $- rac 1m=z+Sm$ 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 $zin mathbb H$, including the vicinity of the singularities.

Edition details

• Publisher: Amer Mathematical Society
• Language: English
• ISBN 10: 1470436833
• ISBN 13: 9781470436834
• Binding: Paperback
• Number of pages: 133