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Integrable structure of Ginibre’s ensemble of real random matrices and a Pfaffian integration theorem. (English) Zbl 1136.82019
Summary: In the recent publication [Phys. Rev. Lett. 95, 230201 (2005), doi:10.1103/PhysRevLett.95.230201)], we reported an exact solution for the probability \(p_{n, k}\) to find exactly \(k\) real eigenvalues in the spectrum of an \(n \times n\) real asymmetric matrix drawn at random from Ginibre’s Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the pfaffian integration theorem, the key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a GinOE matrix restricted to have exactly \(k\) real eigenvalues. In the particular case of \(k =0\), all correlation functions of complex eigenvalues are determined.

MSC:
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
15B52 Random matrices (algebraic aspects)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
58A17 Pfaffian systems
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