Quadratic Vector Equations on Complex Upper Half-Plane
Oskari Ajanki, László Erdős
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.
Rok:
2019
Wydanie:
1
Wydawnictwo:
American Mathematical Society
Język:
english
Strony:
146
ISBN 10:
1470454149
ISBN 13:
9781470454142
Serie:
Memoirs of the American Mathematical Society; 1261
Plik:
PDF, 2.97 MB
IPFS:
,
english, 2019