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

By Laszlo Erdos, Oskari Heikki Ajanki, Torben Kruger

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.

• Format: Paperback
• Language: English
• ISBN: 1470436833
• ISBN13: 9781470436834
• Release Date: December 2019
• Publisher: Amer Mathematical Society
• Length: 133 Pages
• Weight: 0.43 lbs.
• Dimensions: 10.0" x 0.3" x 7.0"