Quadratic vector equations on complex upper half-plane
著者
書誌事項
Quadratic vector equations on complex upper half-plane
（Memoirs of the American Mathematical Society, no. 1261）
American Mathematical Society, c2019
大学図書館所蔵 件 / 全8件
-
該当する所蔵館はありません
- すべての絞り込み条件を解除する
注記
"September 2019, volume 261, number 1261 (fifth of 7 numbers)"
Includes bibliographical reference (p. 131-133)
内容説明・目次
内容説明
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.
目次
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.
「Nielsen BookData」 より
