Progress on the Study of the Ginibre Ensembles / by Sung-Soo Byun, Peter J. Forrester.

Format:

Book

Author/Creator:

Byun, Sung-Soo., Author.

Forrester, Peter J., Author.

Series:

KIAS Springer Series in Mathematics, 2731-5150 ; 3

Language:

English

Subjects (All):

Probabilities.

Mathematical physics.

Differential equations.

Probability Theory.

Mathematical Physics.

Differential Equations.

Local Subjects:

Probability Theory.

Mathematical Physics.

Differential Equations.

Physical Description:

1 online resource (XI, 221 p. 7 illus.)

Edition:

1st ed. 2025.

Place of Publication:

Singapore : Springer Nature Singapore : Imprint: Springer, 2025.

Summary:

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.

Contents:

Introduction

Eigenvalue PDFs and Correlations

Fluctuation Formulas

Coulomb Gas Model, Sum Rules and Asymptotic Behaviours

Normal Matrix Models

Further Theory and Applications

Eigenvalue Statistics for GinOE and Elliptic GinOE

Analogues of GinUE Statistical Properties for GinOE

Further Extensions to GinOE

Statistical Properties of GinSE and Elliptic GinSE

Further Extensions to GinSE.

ISBN:

981-9751-73-X