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

Akemann G, Kanzieper E. Integrable Structure of Ginibre's Ensemble of Real Random Matrices and a Pfaffian Integration Theorem. J. Stat. Phys. 2007;129(5-6):1159-1231.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 theprobability "p_{n,k}" to find exactly "k" real eigenvalues in the spectrum ofan "n" by "n" real asymmetric matrix drawn at random from Ginibre's OrthogonalEnsemble (GinOE). In the present paper, we offer a detailed derivation of theabove result by concentrating on the proof of the Pfaffian integration theorem,the key ingredient of our analysis of the statistics of real eigenvalues in theGinOE. We also initiate a study of the correlations of complex eigenvalues andderive a formula for the joint probability density function of all complexeigenvalues of a GinOE matrix restricted to have exactly "k" real eigenvalues.In the particular case of "k=0", all correlation functions of complexeigenvalues are determined