Quadratic Vector Equations on Complex Upper Half-Plane Paperback / softback
by Oskari Ajanki, Laszlo Erdos, Torben Kruger
Part of the Memoirs of the American Mathematical Society series
Paperback / softback
Description
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.
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Available to Order - This title is available to order, with delivery expected within 2 weeks
- Format:Paperback / softback
- Pages:132 pages
- Publisher:American Mathematical Society
- Publication Date:30/12/2019
- Category:
- ISBN:9781470436834
Other Formats
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Information
-
Available to Order - This title is available to order, with delivery expected within 2 weeks
- Format:Paperback / softback
- Pages:132 pages
- Publisher:American Mathematical Society
- Publication Date:30/12/2019
- Category:
- ISBN:9781470436834
