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Product Code: | MEMO/261/1261 |
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eBook ISBN: | 978-1-4704-5414-2 |
Product Code: | MEMO/261/1261.E |
List Price: | $81.00 |
MAA Member Price: | $72.90 |
AMS Member Price: | $48.60 |
Softcover ISBN: | 978-1-4704-3683-4 |
eBook: ISBN: | 978-1-4704-5414-2 |
Product Code: | MEMO/261/1261.B |
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Softcover ISBN: | 978-1-4704-3683-4 |
Product Code: | MEMO/261/1261 |
List Price: | $81.00 |
MAA Member Price: | $72.90 |
AMS Member Price: | $48.60 |
eBook ISBN: | 978-1-4704-5414-2 |
Product Code: | MEMO/261/1261.E |
List Price: | $81.00 |
MAA Member Price: | $72.90 |
AMS Member Price: | $48.60 |
Softcover ISBN: | 978-1-4704-3683-4 |
eBook ISBN: | 978-1-4704-5414-2 |
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 |
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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.
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Table of Contents
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Chapters
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1. Introduction
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2. Set-up and main results
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3. Local laws for large random matrices
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4. Existence, uniqueness and $\mathrm {L}^{\!2}$-bound
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5. Properties of solution
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6. Uniform bounds
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7. Regularity of solution
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8. Perturbations when generating density is small
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9. Behavior of generating density where it is small
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10. Stability around small minima of generating density
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11. Examples
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A. Appendix
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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. 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
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11. Examples
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A. Appendix