Eigenvalue statistics of the real Ginibre ensemble

TL;DR

This paper analyzes the eigenvalue distributions of real Ginibre matrices using skew orthogonal polynomials, providing explicit correlation formulas and a practical way to compute the largest real eigenvalue's distribution, relevant to biological stability.

Contribution

It introduces explicit Pfaffian formulas for eigenvalue correlations and a tractable method for the largest real eigenvalue distribution in the real Ginibre ensemble.

Findings

Explicit n-point correlation formulas for real and complex eigenvalues.

A computational formula for the largest real eigenvalue distribution.

Relevance to stability analysis in biological networks.

Abstract

The real Ginibre ensemble consists of random $N \times N$ matrices formed from i.i.d. standard Gaussian entries. By using the method of skew orthogonal polynomials, the general $n$-point correlations for the real eigenvalues, and for the complex eigenvalues, are given as $n \times n$ Pfaffians with explicit entries. A computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented. This is relevant to May's stability analysis of biological webs.