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Softcover ISBN:  9781470436834 
Product Code:  MEMO/261/1261 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
eBook ISBN:  9781470454142 
Product Code:  MEMO/261/1261.E 
List Price:  $81.00 
MAA Member Price:  $72.90 
AMS Member Price:  $48.60 
Softcover ISBN:  9781470436834 
eBook ISBN:  9781470454142 
Product Code:  MEMO/261/1261.B 
List Price:  $162.00 $121.50 
MAA Member Price:  $145.80 $109.35 
AMS Member Price:  $97.20 $72.90 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 261; 2019; 133 ppMSC: Primary 45; Secondary 46; 60; 15
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.

Table of Contents

Chapters

1. Introduction

2. Setup 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


Additional Material

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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.

Chapters

1. Introduction

2. Setup 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