Journal of Statistical Physics

, Volume 163, Issue 2, pp 303–323 | Cite as

The Real Ginibre Ensemble with \(k=O(n)\) Real Eigenvalues

  • Luis Carlos García del MolinoEmail author
  • Khashayar Pakdaman
  • Jonathan Touboul
  • Gilles Wainrib


We consider the ensemble of real Ginibre matrices conditioned to have positive fraction \(\alpha >0\) of real eigenvalues. We demonstrate a large deviations principle for the joint eigenvalue density of such matrices and introduce a two phase log-gas whose stationary distribution coincides with the spectral measure of the ensemble. Using these tools we provide an asymptotic expansion for the probability \(p^n_{\alpha n}\) that an \(n\times n\) Ginibre matrix has \(k=\alpha n\) real eigenvalues and we characterize the spectral measures of these matrices.


Real Ginibre matrices Large deviations Log-gas 



We thank an anonymous referee for his suggestions on the proof of Theorem 1.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Luis Carlos García del Molino
    • 1
    • 2
    Email author
  • Khashayar Pakdaman
    • 1
  • Jonathan Touboul
    • 2
    • 3
  • Gilles Wainrib
    • 4
  1. 1.Institut Jacques Monod, CNRS UMR 7592Université Paris Diderot, Paris Cité SorbonneParisFrance
  2. 2.Mathematical neuroscience TeamCIRB-Collège de France (CNRS UMR 7241, INSERM U1050, UPMC ED 158, MEMOLIFE PSL)ParisFrance
  3. 3.MYCENAE TeamINRIA Paris-RocquencourtParisFrance
  4. 4.Département d’Informatique (DATA)Ecole Normale SupérieureParisFrance

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