Quadratic Vector Equations On Complex Upper Half Plane
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Quadratic Vector Equations on Complex Upper Half-plane
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.
Quadratic Vector Equations on Complex Upper Half-Plane
Author: Oskari Ajanki
language: en
Publisher: American Mathematical Soc.
Release Date: 2019-12-02
The authors consider the nonlinear equation −1m=z+Sm with a parameter z in the complex upper half plane H, where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on 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∈H, including the vicinity of the singularities.
Random Matrices
Author: Alexei Borodin
language: en
Publisher: American Mathematical Soc.
Release Date: 2019-10-30
Random matrix theory has many roots and many branches in mathematics, statistics, physics, computer science, data science, numerical analysis, biology, ecology, engineering, and operations research. This book provides a snippet of this vast domain of study, with a particular focus on the notations of universality and integrability. Universality shows that many systems behave the same way in their large scale limit, while integrability provides a route to describe the nature of those universal limits. Many of the ten contributed chapters address these themes, while others touch on applications of tools and results from random matrix theory. This book is appropriate for graduate students and researchers interested in learning techniques and results in random matrix theory from different perspectives and viewpoints. It also captures a moment in the evolution of the theory, when the previous decade brought major break-throughs, prompting exciting new directions of research.