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Quadratic Vector Equations On Complex Upper Half-Plane
About this Title
Oskari Heikki Ajanki, Institute of Science and Technology Austria, László Erdős, Institute of Science and Technology Austria and Torben Krüger, Institute of Science and Technology Austria
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 261, Number 1261
ISBNs: 978-1-4704-3683-4 (print); 978-1-4704-5414-2 (online)
DOI: https://doi.org/10.1090/memo/1261
Published electronically: November 13, 2019
Keywords: Stieltjes-transform,
algebraic singularity,
density of states,
cubic cusp,
random matrices,
Wigner-type random matrix,
local law
MSC: Primary 45Gxx; Secondary 46Txx, 60B20, 15B52
Table of Contents
Chapters
- 1. Introduction
- 2. Set-up and main results
- 3. Local laws for large random matrices
- 4. Existence, uniqueness and $\mathrm {L}^{\!2}$-bound
- 5. Properties of solution
- 6. Uniform bounds
- 7. Regularity of solution
- 8. Perturbations when generating density is small
- 9. Behavior of generating density where it is small
- 10. Stability around small minima of generating density
- 11. Examples
- A. Appendix
Abstract
We consider the nonlinear equation $-\frac {1}{m}=z+Sm$ 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 Ajanki, Erdős, and Krüger (2016b), we 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 we give a comprehensive analysis of these singularities with uniform quantitative controls. We 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 companion paper, Ajanki, Erdős, and Krüger (2016c), we present a complete stability analysis of the equation for any $z\in \mathbb {H}$, including the vicinity of the singularities.- Johannes Alt, Singularities of the density of states of random Gram matrices, Electron. Commun. Probab. 22 (2017), Paper No. 63, 13. MR 3734102, DOI 10.1214/17-ECP97
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