# Dyson equation and eigenvalue statistics of random matrices (Record no. 373573)

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000 -LEADER | |
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fixed length control field | 03467ntm a22003857a 4500 |

003 - CONTROL NUMBER IDENTIFIER | |

control field | AT-ISTA |

005 - DATE AND TIME OF LATEST TRANSACTION | |

control field | 20190813150243.0 |

008 - FIXED-LENGTH DATA ELEMENTS--GENERAL INFORMATION | |

fixed length control field | 180813s2018 au ||||| m||| 00| 0 eng d |

040 ## - CATALOGING SOURCE | |

Transcribing agency | IST |

100 ## - MAIN ENTRY--PERSONAL NAME | |

Personal name | Alt, Johannes |

9 (RLIN) | 4395 |

245 ## - TITLE STATEMENT | |

Title | Dyson equation and eigenvalue statistics of random matrices |

260 ## - PUBLICATION, DISTRIBUTION, ETC. (IMPRINT) | |

Name of publisher, distributor, etc. | IST Austria |

Date of publication, distribution, etc. | 2018 |

500 ## - GENERAL NOTE | |

General note | Thesis |

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Formatted contents note | Biographical Sketch |

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Formatted contents note | List of Publications |

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Formatted contents note | Acknowledgments |

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Formatted contents note | Abstract |

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Formatted contents note | List of Tables |

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Formatted contents note | List of Figures |

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Formatted contents note | List of Symbols |

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Formatted contents note | List of Abbreviations |

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Formatted contents note | Chapter 1. Introduction |

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Formatted contents note | Chapter 2. Overview of the results |

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Formatted contents note | Chapter 3. The local semicircle law for random matrices with a fourfold symmetry |

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Formatted contents note | Chapter 4. Local law for random Gram matrices |

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Formatted contents note | Chapter 5. Singularities of the density of states of random Gram matrices |

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Formatted contents note | Chapter 6. Local inhomogeneous circular law |

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Formatted contents note | Chapter 7. Location of the spectrum of Kronecker random matrices |

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Formatted contents note | Chapter 8. The Dyson equation with linear self-energy: spectral bands, edges and cusps |

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Formatted contents note | Chapter 9. Correlated Random Matrices: Band Rigidity and Edge Universality |

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Formatted contents note | Bibliography |

520 ## - SUMMARY, ETC. | |

Summary, etc. | The eigenvalue density of many large random matrices is well approximated by a deterministic measure, the self-consistent density of states. In the present work, we show this behaviour for several classes of random matrices. In fact, we establish that, in each of these classes, the self-consistent density of states approximates the eigenvalue density of the random matrix on all scales slightly above the typical eigenvalue spacing. For large classes of random matrices, the self-consistent density of states exhibits several universal features. We prove that, under suitable assumptions, random Gram matrices and Hermitian random matrices with decaying correlations have a 1/3-Hölder continuous self-consistent density of states ρ on R, which is analytic, where it is positive, and has either a square root edge or a cubic root cusp, where it vanishes. We, thus, extend the validity of the corresponding result for Wigner-type matrices from [4, 5, 7]. We show that ρ is determined as the inverse Stieltjes transform of the normalized trace of the unique solution m(z) to the Dyson equation −m(z) −1 = z − a + S[m(z)] on C N×N with the constraint Im m(z) ≥ 0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of C N×N and S is a positivity-preserving operator on C N×N encoding the first two moments of the random matrix. In order to analyze a possible limit of ρ for N → ∞ and address some applications in free probability theory, we also consider the Dyson equation on infinite dimensional von Neumann algebras. We present two applications to random matrices. We first establish that, under certain assumptions, large random matrices with independent entries have a rotationally symmetric self-consistent density of states which is supported on a centered disk in C. Moreover, it is infinitely often differentiable apart from a jump on the boundary of this disk. Second, we show edge universality at all regular (not necessarily extreme) spectral edges for Hermitian random matrices with decaying correlations. |

856 ## - ELECTRONIC LOCATION AND ACCESS | |

Uniform Resource Identifier | https://doi.org/10.15479/AT:ISTA:TH_1040 |

942 ## - ADDED ENTRY ELEMENTS (KOHA) | |

Source of classification or shelving scheme |

Withdrawn status | Lost status | Source of classification or shelving scheme | Damaged status | Not for loan | Permanent Location | Current Location | Date acquired | Barcode | Date last seen | Price effective from | Koha item type |
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Not Lost | Library | Library | 2018-08-13 | AT-ISTA#001868 | 2018-08-13 | 2018-08-13 | Book |