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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1 | (4) |
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Chapter 2 Set-up and main results |
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5 | (16) |
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7 | (6) |
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13 | (3) |
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2.3 Relationship between Theorem 2.6 and Theorem 2.6 of [ AEK16b] |
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16 | (1) |
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17 | (4) |
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Chapter 3 Local laws for large random matrices |
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21 | (6) |
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3.1 Proof of local law inside bulk of the spectrum |
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23 | (4) |
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Chapter 4 Existence, uniqueness and L2-bound |
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27 | (8) |
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4.1 Stieltjes transform representation |
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30 | (2) |
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4.2 Operator F and structural L2-bound |
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32 | (3) |
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Chapter 5 Properties of solution |
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35 | (12) |
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5.1 Relations between components of m and F |
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35 | (6) |
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5.2 Stability and operator B |
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41 | (6) |
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47 | (8) |
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6.1 Uniform bounds from L2-estimates |
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49 | (2) |
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6.2 Uniform bound around z = 0 when a = 0 |
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51 | (4) |
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Chapter 7 Regularity of solution |
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55 | (8) |
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Chapter 8 Perturbations when generating density is small |
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63 | (10) |
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8.1 Expansion of operator B |
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64 | (4) |
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68 | (5) |
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Chapter 9 Behavior of generating density where it is small |
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73 | (28) |
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9.1 Expansion around non-zero minima of generating density |
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77 | (4) |
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9.2 Expansions around minima where generating density vanishes |
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81 | (15) |
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9.3 Proofs of Theorems 2.6 and 2.11 |
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96 | (5) |
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Chapter 10 Stability around small minima of generating density |
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101 | (6) |
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107 | (10) |
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11.1 The band operator, lack of self-averaging, and property A3 |
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108 | (1) |
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11.2 Divergences in SB, outliers, and function T |
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108 | (4) |
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11.3 Blow-up at z = 0 when a = 0 and assumption B1 |
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112 | (1) |
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11.4 Effects of non-constant function a |
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113 | (1) |
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11.5 Discretization and reduction of the QVE |
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113 | (2) |
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11.6 Simple example that exhibits all universal shapes |
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115 | (2) |
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117 | (14) |
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A.1 Proofs of auxiliary results in
Chapter 4 |
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118 | (1) |
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A.2 Proofs of auxiliary results in
Chapter 5 |
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119 | (1) |
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A.3 Scalability of matrices with non-negative entries |
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120 | (3) |
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A.4 Variational bounds when Re z = 0 |
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123 | (3) |
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A.5 Holder continuity of Stieltjes transform |
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126 | (2) |
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A.6 Cubic roots and associated auxiliary functions |
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128 | (3) |
| Bibliography |
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131 | |
Oskari Ajanki, Institute of Science and Technology, Klosterneuberg, Austria.
Laszlo Erdos, Institute of Science and Technology, Klosterneuberg, Austria.
Torben Kruger, Institute of Science and Technology, Klosterneuberg, Austria.
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