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

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematical Sciences and BURSt Research Centre, Brunel University West London, Uxbridge, UB8 3PH, UK

    Gernot Akemann

  2. Department of Applied Mathematics, H.I.T.—Holon Institute of Technology, Holon, 58102, Israel

    Eugene Kanzieper

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  1. Gernot Akemann
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  2. Eugene Kanzieper
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Correspondence to Eugene Kanzieper.

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Akemann, G., Kanzieper, E. Integrable Structure of Ginibre’s Ensemble of Real Random Matrices and a Pfaffian Integration Theorem. J Stat Phys 129, 1159–1231 (2007). https://doi.org/10.1007/s10955-007-9381-2

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