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Progress on the Study of the Ginibre Ensembles

  • Book
  • Open Access
  • Sep 2024

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Overview

  • This book is open access, which means that you have free and unlimited access
  • Is the first book that focuses on the Ginibre ensembles
  • Presents the subject relevant to a broad range of researchers
  • Suits for self-study, as well as for reference purposes, making it suitable for graduate students

Part of the book series: KIAS Springer Series in Mathematics (KIASSSM, volume 3)

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About this book

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.

Keywords

  • Open Access
  • Ginibre Ensembles
  • Non-Hermitian Random Matrices
  • Determinantal Point Processes
  • Pfaffan Point Processes
  • Orthogonal Polynomials in the Complex Plane
  • Skew Orthogonal Polynomials
  • Two-Dimensional Coulomb Gas
  • Normal Matrix Model
  • Fluctuation Formulas

Authors and Affiliations

  • Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, Korea (Republic of)

    Sung-Soo Byun

  • School of Mathematics and Statistics, University of Melbourne, Melbourne , Australia

    Peter J. Forrester

About the authors

Sung-Soo Byun is Assistant Professor in the Department of Mathematical Sciences at Seoul National University. 
Peter J. Forrester is Professor in School of Mathematics and Statistics at The University of Melbourne.

Bibliographic Information

  • Book Title: Progress on the Study of the Ginibre Ensembles

  • Authors: Sung-Soo Byun, Peter J. Forrester

  • Series Title: KIAS Springer Series in Mathematics

  • Publisher: Springer Singapore

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s) 2024

  • Hardcover ISBN: 978-981-97-5172-3Due: 30 September 2024

  • Softcover ISBN: 978-981-97-5175-4Due: 30 September 2025

  • eBook ISBN: 978-981-97-5173-0Due: 30 September 2024

  • Series ISSN: 2731-5142

  • Series E-ISSN: 2731-5150

  • Edition Number: 1

  • Number of Pages: X, 214

  • Number of Illustrations: 6 b/w illustrations

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