Eigenvalue statistics of the real Ginibre ensemble

Abstract

The real Ginibre ensemble consists of random N×N matrices formed from independent and identically distributed 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×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. © 2007 The American Physical Society.