Statistics of Real Eigenvalues in Ginibre's Ensemble of Random Real Matrices
Kanzieper, Eugene · Akemann, Gernot
Mathematical Physics
Disordered Systems and Neural Networks
High Energy Physics - Theory
Exactly Solvable and Integrable Systems
Original · EN
The integrable structure of Ginibre's Orthogonal Ensemble of random matrices is looked at through the prism of the probability "pₙ,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 "pₙ,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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