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Dyson equation and eigenvalue statistics of random matrices

By: Alt, Johannes.
Material type: materialTypeLabelBookPublisher: IST AUSTRIA 2018
Contents:
Biographical Sketch List of Publications Acknowledgments Abstract List of Tables List of Figures List of Symbols List of Abbreviations Chapter 1. Introduction Chapter 2. Overview of the results 2.1. Outlook Chapter 3. The local semicircle law for random matrices with a fourfold symmetry 3.1. Introduction 3.2. Main Result 3.3. Fourier Transform of Random Matrices 3.4. Tools 3.5. Proof of the Main Result 3.6. Proof of the Fluctuation Averaging Chapter 4. Local law for random Gram matrices 4.1. Introduction 4.2. Main results 4.3. Quadratic vector equation 4.4. Local laws 4.5. Proof of the Rotation-Inversion lemma Chapter 5. Singularities of the density of states of random Gram matrices 5.1. Introduction 5.2. Main results 5.3. Quadratic vector equation 5.4. Proofs of Theorem 5.2.3 and Theorem 5.2.6 Chapter 6. Local inhomogeneous circular law 6.1. Introduction 6.2. Main results 6.3. Dyson equation for the inhomogeneous circular law 6.4. Proof of Proposition 6.2.5 6.5. Local law 6.6. Proof of Lemma 6.2.3 6.7. Proof of the Contraction-Inversion Lemma Chapter 7. Location of the spectrum of Kronecker random matrices 7.1. Introduction 7.2. Main results 7.3. Solution and stability of the Dyson equation 7.4. Hermitian Kronecker matrices 7.5. Fluctuation Averaging: Proof of Proposition 7.4.6 7.6. Non-Hermitian Kronecker matrices and proof of Theorem 7.2.4 7.7. An alternative definition of the self-consistent ε-pseudospectrum 7.8. Proofs of Theorem 7.2.7 and Lemma 7.4.8 Chapter 8. The Dyson equation with linear self-energy: spectral bands, edges and cusps 8.1. Introduction 8.2. Main results 8.3. The solution of the Dyson equation 8.4. Regularity of the solution and the density of states 8.5. Spectral properties of the stability operator for small self-consistent density of states 8.6. The cubic equation 8.7. Cubic analysis 8.8. Band mass formula – Proof of Proposition 8.2.6 8.9. Dyson equation for Kronecker random matrices 8.10. Perturbations of the data pair 8.11. Stieltjes transforms of positive operator-valued measures 8.12. Positivity-preserving, symmetric operators on A 8.13. Non-Hermitian perturbation theory 8.14. Characterization of supp ρ Chapter 9. Correlated Random Matrices: Band Rigidity and Edge Universality 9.1. Introduction 9.2. Main results 9.3. Proof of the local law 9.4. Proof of Universality 9.5. Auxiliary results Bibliography
List(s) this item appears in: IST Austria Thesis 2018
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Thesis

Biographical Sketch
List of Publications
Acknowledgments
Abstract
List of Tables
List of Figures
List of Symbols
List of Abbreviations
Chapter 1. Introduction
Chapter 2. Overview of the results
2.1. Outlook
Chapter 3. The local semicircle law for random matrices with a fourfold symmetry
3.1. Introduction
3.2. Main Result
3.3. Fourier Transform of Random Matrices
3.4. Tools
3.5. Proof of the Main Result
3.6. Proof of the Fluctuation Averaging
Chapter 4. Local law for random Gram matrices
4.1. Introduction
4.2. Main results
4.3. Quadratic vector equation
4.4. Local laws
4.5. Proof of the Rotation-Inversion lemma
Chapter 5. Singularities of the density of states of random Gram matrices
5.1. Introduction
5.2. Main results
5.3. Quadratic vector equation
5.4. Proofs of Theorem 5.2.3 and Theorem 5.2.6
Chapter 6. Local inhomogeneous circular law
6.1. Introduction
6.2. Main results
6.3. Dyson equation for the inhomogeneous circular law
6.4. Proof of Proposition 6.2.5
6.5. Local law
6.6. Proof of Lemma 6.2.3
6.7. Proof of the Contraction-Inversion Lemma
Chapter 7. Location of the spectrum of Kronecker random matrices
7.1. Introduction
7.2. Main results
7.3. Solution and stability of the Dyson equation
7.4. Hermitian Kronecker matrices
7.5. Fluctuation Averaging: Proof of Proposition 7.4.6
7.6. Non-Hermitian Kronecker matrices and proof of Theorem 7.2.4
7.7. An alternative definition of the self-consistent ε-pseudospectrum
7.8. Proofs of Theorem 7.2.7 and Lemma 7.4.8
Chapter 8. The Dyson equation with linear self-energy: spectral bands, edges and cusps
8.1. Introduction
8.2. Main results
8.3. The solution of the Dyson equation
8.4. Regularity of the solution and the density of states
8.5. Spectral properties of the stability operator for small self-consistent density of states
8.6. The cubic equation
8.7. Cubic analysis
8.8. Band mass formula – Proof of Proposition 8.2.6
8.9. Dyson equation for Kronecker random matrices
8.10. Perturbations of the data pair
8.11. Stieltjes transforms of positive operator-valued measures
8.12. Positivity-preserving, symmetric operators on A
8.13. Non-Hermitian perturbation theory
8.14. Characterization of supp ρ
Chapter 9. Correlated Random Matrices: Band Rigidity and Edge Universality
9.1. Introduction
9.2. Main results
9.3. Proof of the local law
9.4. Proof of Universality
9.5. Auxiliary results
Bibliography

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