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

Authors:
  • EKEugene Kanzieper
    Eugene Kanzieper
  • GAGernot Akemann
    Gernot Akemann
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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.

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