Advertisement

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
Article

Abstract

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.

Keywords

Real Ginibre matrices Large deviations Log-gas 

Notes

Acknowledgments

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

References

  1. 1.
    Forrester, P.J.: London Mathematical Society Monographs. Princeton University Press, Princeton (2010)Google Scholar
  2. 2.
    Tao, T.: Topics in Ramdom Matrix Theory. American Mathematical Society, Providence (2012)CrossRefGoogle Scholar
  3. 3.
    Bordenave, C., Chafai, D.: Around the circular law. Probab. Surv. 93, 1–89 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Wigner, E.P.: Lower limit for the energy derivative of the scattering phase shift. Phys. Rev. 98(1), 145–147 (1955)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Auffinger, A., Ben Arous, G., Černỳ, J.: Random matrices and complexity of spin glasses. Commun. Pure Appl. Math. 66(2), 165–201 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    May, R.M.: Will a large complex system be stable? Nature 238, 413–414 (1972)ADSCrossRefGoogle Scholar
  7. 7.
    Wainrib, G., Touboul, J.: Topological and dynamical complexity of random neural networks. Phys. Rev. Lett. 110(11), 118101 (2013)ADSCrossRefGoogle Scholar
  8. 8.
    del Molino, L.C.G., Pakdaman, K., Touboul, J., Wainrib, G.: Synchronization in random balanced networks. Phys. Rev. E 88(4), 042824 (2013)ADSCrossRefGoogle Scholar
  9. 9.
    Couillet, R., Debbah, M., et al.: Random Matrix Methods for Wireless Communications. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  10. 10.
    Jaeger, H., Haas, H.: Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication. Science 304(5667), 78–80 (2004)ADSCrossRefGoogle Scholar
  11. 11.
    Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lehmann, N., Sommers, H.J.: Eigenvalue statistics of random real matrices. Phys. Rev. Lett. 67, 941–944 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Edelman, A.: The probability that a random real Gaussian matrix has k real Eigenvalues, related distributions, and the circular law. J. Multivar. Anal. 60, 203–232 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Forrester, P.J., Nagao, T.: Eigenvalue statistics of the real Ginibre enesmble. Phys. Rev. Lett. 99, 050603 (2007)ADSCrossRefGoogle Scholar
  15. 15.
    Sommers, H.J.: Symplectic structure of the real Ginibre ensemble. J. Phys. A 40(29), F671 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Forrester, P.J., Nagao, T.: Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble. J. Phys. A 41(37), 375003 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Forrester, P.J., Mays, A.: A method to calculate correlation functions for \(\beta =1\) random matrices of odd size. J. Phys. A 134(3), 443–462 (2009)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Sinclair, C.: Correlation functions for \(\beta =1\) ensembles of matrices of odd size. J. Stat. Phys. 136(1), 17–33 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sommers, H.J., Wieczorek, W.: General eigenvalue correlations for the real Ginibre ensemble. J. Phys. A 41(40), 405003 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Borodin, A., Sinclair, C.: The Ginibre ensemble of real random matrices and its scaling limits. Commun. Math. Phys. 291(1), 177–224 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rider, B., Sinclair, C.D., et al.: Extremal laws for the real ginibre ensemble. Ann. Appl. Probab. 24(4), 1621–1651 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tao, T., Vu, V.: Random matrices: universality of ESD and the circular law. Ann. Probab. 38(5), 2023–2065 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tao, T., Vu, V.: Random matrices: universality of local spectral statistics of non-hermitian matrices (2012). arXiv:1206.1893
  24. 24.
    Bourgade, P., Erdos, L., Yau, H.: Universality of general \( beta \)-ensembles. Duke Mathematical Journal 163(6), 1127–1190 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Bourgade P., Erdos, L., Yau, H., Yin, J.: Fixed energy universality for generalized Wigner matrices (2014). arXiv:1407.5606
  26. 26.
    Erdös, L., Yau, H.-T., Yin, J.: Rigidity of eigenvalues of generalized wigner matrices. Adv. Math. 229(3), 1435–1515 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Bourgade, P., Erdös, L., Yau, H.-T.: Bulk universality of general \(\beta \)-ensembles with non-convex potential. J. Math. Phys. 53(9), 095221 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Bourgade, P., Erdös, L., Yau, H.-T.: Edge universality of beta ensembles. Commun. Math. Phys. 332(1), 261–353 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Bourgade, P., Yau, H.-T., Yin, J.: Local circular law for random matrices. Probab. Theory Relat. Fields 159(3–4), 545–595 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Bourgade, P., Yau, H.-T., Yin, J.: The local circular law ii: the edge case. Probab. Theory Relat. Fields 159(3–4), 619–660 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Ben Arous, G., Guionnet, A.: Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Relat. Fields 108, 517–542 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ben Arous, G., Zeitouni, O.: Large Deviations from the circular law. ESAIM: Probab. Stat. 2, 123–134 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Edelman, A., Kostlan, E., Shub, M.: How many Eigenvalues of a random matrix are real? J. Am. Math. Soc. 7, 247–267 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ginibre, J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449 (1965)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Sommers, H.J., Crisanti, A., Sompolinsky, H.: Spectrum of large random asymmetric matrices. Phys. Rev. Lett. 60, 1895 (1988)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Tribe, R., Zaboronski, O.: Pfaffian formulae for one dimensional coalescing and annihilating systems. Electron. J. Probab. 163(76), 2080–2103 (2011)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Forrester, P.J.: Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble. arXiv:1306.4106
  38. 38.
    Beenakker, C.: Random-matrix theory of majorana fermions and topological superconductors (2014). arXiv:1407.2131
  39. 39.
    Kanzieper, E., Akemann, G.: Statistics of real eigenvalues in Ginibre’s ensemble of random real matrices. Phys. Rev. Lett. 95(23), 230201 (2005)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Akemann, G., Kanzieper, E.: Integrable structure of Ginibre’s ensemble of real random matrices and a Pfaffian integration theorem. J. Stat. Phys. 129(5–6), 1159–1231 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Kanzieper, E., Poplavskyi, M., Timm, C., Tribe, R., Zaboronski, O.: What is the probability that a large random matrix has no real eigenvalues? (2015). arXiv:1503.07926
  42. 42.
    Dyson, F.J.: A Brownian-motion model for the Eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198 (1962)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Majumdar, S.N., Nadal, C., Scardicchio, A., Vivo, P.: Index distribution of gaussian random matrices. Phys. Rev. Lett. 103(22), 220603 (2009)ADSCrossRefGoogle Scholar
  44. 44.
    Majumdar, S.N., Vivo, P.: Number of relevant directions in principal component analysis and wishart random matrices. Phys. Rev. Lett. 108(20), 200601 (2012)ADSCrossRefGoogle Scholar
  45. 45.
    Rogers, L., Shi, Z.: Interacting Brownian particles and the Wigner law. Probab. Theory Relat. Fields 95, 555–570 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Cepa, E., Lepingle, D.: Diffusing particles with electrostatic repulsion. Probab. Theory Relat. Fields 107, 429–449 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Anderson, W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  48. 48.
    Sandier, E., Serfaty, S.: 1d log gases and the renormalized energy: crystallization at vanishing temperature. Probab. Theory Relat Fields 162, 1–52 (2014)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Sandier, E., Serfaty, S.: 2d coulomb gases and the renormalized energy (2012). arXiv:1201.3503
  50. 50.
    Rougerie, N., Serfaty, S.: Higher dimensional coulomb gases and renormalized energy functionals (2013). arXiv:1307.2805
  51. 51.
    Allez, R., Touboul, J., Wainrib, G.: Index distribution of the Ginibre ensemble. J. Phys. A 47(4), 042001 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Braides, A.: Gamma-convergence for Beginners. Oxford University Press, Oxford (2002)CrossRefzbMATHGoogle Scholar
  53. 53.
    Armstrong, S.N., Serfaty, S., Zeitouni, O.: Remarks on a constrained optimization problem for the ginibre ensemble. Potential Anal. 41(3), 945–958 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Abrikosov, A.A.: On the magnetic properties of superconductors of the second type. Sov. Phys. JETP 5, 1174–1182 (1957)Google Scholar
  55. 55.
    Chafaï, D., Gozlan, N., Zitt, P.-A.: First order global asymptotics for confined particles with singular pair repulsion (2013). arXiv:1304.7569
  56. 56.
    del Molino, L.C.G., Pakdaman, K., Touboul, J.: The heterogeneous gas with singular interaction: generalized circular law and heterogeneous renormalized energy. J. Phys. A 48(4), 045208 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Vivo, P., Majumdar, S.N., Bohigas, O.: Large deviations and random matrices. Acta Phys. Pol. B 38(13), 4139 (2007)ADSMathSciNetzbMATHGoogle Scholar

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

Personalised recommendations