Integrable Structure of Ginibre's Ensemble of Real Random Matrices and a Pfaffian Integration Theorem

TL;DR

This paper provides a detailed derivation of the probability distribution of real eigenvalues in Ginibre's real ensemble, introduces a Pfaffian integration theorem, and studies correlations of complex eigenvalues.

Contribution

It offers a rigorous proof of the Pfaffian integration theorem and extends understanding of eigenvalue statistics in GinOE matrices.

Findings

Derived the probability 'p_{n,k}' for exactly 'k' real eigenvalues.

Formulated the joint probability density function for complex eigenvalues with 'k' real eigenvalues.

Determined all correlation functions for complex eigenvalues when 'k=0'.

Abstract

In the recent publication [E. Kanzieper and G. Akemann, Phys. Rev. Lett. 95, 230201 (2005); arXiv: math-ph/0507058], an exact solution was reported for the probability "p_{n,k}" to find exactly "k" real eigenvalues in the spectrum of an "n" by "n" real asymmetric matrix drawn at random from Ginibre's Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a GinOE matrix restricted to have exactly "k" real eigenvalues. In the particular case of "k=0", all correlation functions of complex eigenvalues are determined.