Statistics of Real Eigenvalues in Ginibre's Ensemble of Random Real Matrices

TL;DR

This paper derives an exact formula for the probability of finding a specific number of real eigenvalues in Ginibre's real random matrices, revealing a deep connection to symmetric functions and extending Dyson's theorem.

Contribution

It provides the first exact solution for the distribution of real eigenvalues in Ginibre's real ensemble, linking it to symmetric function theory and extending Dyson's integration theorem.

Findings

Exact probability formula for real eigenvalues

Connection to symmetric functions theory

Extension of Dyson's integration theorem

Abstract

The integrable structure of Ginibre's Orthogonal Ensemble of random matrices is looked at through the prism of the probability "p_{n,k}" to find exactly "k" real eigenvalues in the spectrum of an "n" by "n" real asymmetric Gaussian random matrix. The exact solution for the probability function "p_{n,k}" is presented, and its remarkable connection to the theory of symmetric functions is revealed. An extension of the Dyson integration theorem is a key ingredient of the theory presented.