Progress on the study of the Ginibre ensembles II: GinOE and GinSE

TL;DR

This paper reviews recent advances in the theory of real and quaternion Ginibre ensembles, focusing on eigenvalue distributions, Pfaffian point processes, and applications in physics and mathematics.

Contribution

It provides a comprehensive account of the eigenvalue statistics and new theoretical developments for GinOE and GinSE, extending the complex case analysis.

Findings

Eigenvalue correlation functions and limit formulas are now well-understood.

Development of skew orthogonal polynomials for Pfaffian processes.

Applications include diffusion processes, quantum dots, and nonlinear differential equations.

Abstract

This is part II of a review relating to the three classes of random non-Hermitian Gaussian matrices introduced by Ginibre in 1965. While part I restricted attention to the GinUE (Ginibre unitary ensemble) case of complex elements, in this part the cases of real elements (GinOE, denoting Ginibre orthogonal ensemble) and quaternion elements represented as $2 \times 2$ complex blocks (GinSE, denoting Ginibre symplectic ensemble) are considered. The eigenvalues of both GinOE and GinSE form Pfaffian point processes, which are more complicated than the determinantal point processes resulting from GinUE. Nevertheless, many of the obstacles that have slowed progress on the development of traditional aspects of the theory have now been overcome, while new theoretical aspects and new applications have been identified. This permits a comprehensive account of themes addressed too in the complex case: eigenvalue probability density functions and correlation functions, limit formulas for correlation functions, fluctuation formulas, sum rules, gap probabilities and eigenvector statistics, among others. Distinct from the complex case is the need to develop a theory of skew orthogonal polynomials corresponding to the skew inner product associated with the Pfaffian. Another distinct theme is the statistics of real eigenvalues, which is unique to GinOE. These appear in a number of applications of the theory, coming from areas as diverse as diffusion processes and persistence in statistical physics, topologically driven parametric energy level crossings for certain quantum dots, and equilibria counting for a system of random nonlinear differential equations.