Quadratic Vector Equations on Complex Upper Half-Plane
Seiten
2019
American Mathematical Society (Verlag)
978-1-4704-3683-4 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-3683-4 (ISBN)
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Considers 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 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.
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.
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.
Introduction
Set-up and main results
Local laws for large random matrices
Existence, uniqueness and $/mathrm{L}^{2}$-bound
Properties of solution
Uniform bounds
Regularity of solution
Perturbations when generating density is small
Behavior of generating density where it is small
Stability around small minima of generating density
Examples
Appendix A.
Bibliography.
| Erscheinungsdatum | 31.12.2019 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 280 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| ISBN-10 | 1-4704-3683-3 / 1470436833 |
| ISBN-13 | 978-1-4704-3683-4 / 9781470436834 |
| Zustand | Neuware |
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