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Progress on the Study of the Ginibre Ensembles
Sung-Soo Byun and Peter J. Forrester
2025
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This open access book focuses on the Ginibre ensembles that are non-Hermitian random matrices proposed by Ginibre in 1965. Since that time, they have enjoyed prominence within random matrix theory, featuring, for example, the first book on the subject written by Mehta in 1967. Their status has been consolidated and extended over the following years, as more applications have come to light, and the theory has developed to greater depths. This book sets about detailing much of this progress. Themes covered include eigenvalue PDFs and correlation functions, fluctuation formulas, sum rules and asymptotic behaviors, normal matrix models, and applications to quantum many-body problems and quantum chaos. There is a distinction between the Ginibre ensemble with complex entries (GinUE) and those with real or quaternion entries (GinOE and GinSE, respectively). First, the eigenvalues of GinUE form a determinantal point process, while those of GinOE and GinSE have the more complicated structure of a Pfaffian point process. Eigenvalues on the real line in the case of GinOE also provide another distinction. On the other hand, the increased complexity provides new opportunities for research. This is demonstrated in our presentation, which details several applications and contains not previously published theoretical advances. The areas of application are diverse, with examples being diffusion processes and persistence in statistical physics and equilibria counting for a system of random nonlinear differential equations in the study of the stability of complex systems.
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Keywords
- Determinantal Point Processes
- Fluctuation Formulas
- Ginibre Ensembles
- Non-Hermitian Random Matrices
- Normal Matrix Model
- Orthogonal Polynomials in the Complex Plane
- Pfaffan Point Processes
- Skew Orthogonal Polynomials
- thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKJ Differential calculus and equations
- thema EDItEUR::P Mathematics and Science::PB Mathematics::PBT Probability and statistics
- thema EDItEUR::P Mathematics and Science::PB Mathematics::PBW Applied mathematics::PBWL Stochastics
- thema EDItEUR::P Mathematics and Science::PH Physics::PHU Mathematical physics
- Two-Dimensional Coulomb Gas
Links
DOI: 10.1007/978-981-97-5173-0web: https://link.springer.com/book/10.1007/978-981-97-5173-0
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