Journal of Statistical Physics

, Volume 129, Issue 5–6, pp 1159–1231 | Cite as

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

  • Gernot Akemann
  • Eugene KanzieperEmail author


In the recent publication (E. Kanzieper and G. Akemann in Phys. Rev. Lett. 95:230201, 2005), an exact solution was reported for the probability p n,k to find exactly k real eigenvalues in the spectrum of an n×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.


Random Matrix Stat Phys Real Eigenvalue Projection Property Random Matrix Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mathematical Sciences and BURSt Research CentreBrunel University West LondonUxbridgeUK
  2. 2.Department of Applied MathematicsH.I.T.—Holon Institute of TechnologyHolonIsrael

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