# Dyson equation and eigenvalue statistics of random matrices

##### By: Alt, Johannes.

Material type: BookPublisher: IST AUSTRIA 2018Item type | Current location | Call number | Status | Date due | Barcode | Item holds |
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Book | Library | Available |

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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