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

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 $n\times n$ 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.