Journal of Statistical Physics

, Volume 161, Issue 3, pp 633–656 | Cite as

Poisson Statistics for Matrix Ensembles at Large Temperature

Article

Abstract

In this article, we consider \(\beta \)-ensembles, i.e. collections of particles with random positions on the real line having joint distribution
$$\begin{aligned} \frac{1}{Z_N(\beta )}\big |\Delta (\lambda )\big |^\beta e^{- \frac{N\beta }{4}\sum _{i=1}^N\lambda _i^2}\mathrm {d}\lambda , \end{aligned}$$
in the regime where \(\beta \rightarrow 0\) as \(N\rightarrow \infty \). We briefly describe the global regime and then consider the local regime. In the case where \(N\beta \) stays bounded, we prove that the local eigenvalue statistics, in the vicinity of any real number, are asymptotically to those of a Poisson point process. In the case where \(N\beta \rightarrow \infty \), we prove a partial result in this direction.

Keywords

Random matrices \(\beta \)-Ensembles Poisson point process 

Mathematics Subject Classification

15A52 60F05 

References

  1. 1.
    Allez, R., Guionnet, A.: A diffusive model for invariant \(\beta \)-ensembles. Electron. J. Probab. 18(62), 1–30 (2013)MathSciNetGoogle Scholar
  2. 2.
    Allez, R., Bouchaud, J.-P., Guionnet, A.: Invariant beta ensembles and the Gauss-Wigner crossover. Phys. Rev. Lett. 109(9), 094–102 (2013)Google Scholar
  3. 3.
    Allez, R., Bouchaud, J.-P., Majumdar, S.N., Vivo, P.: Invariant\(\beta \)-Wishart ensembles, crossover densities and asymptotic corrections to the Marcenko-Pastur law. J. Phys. 46(1), 015001 (2013)MathSciNetADSGoogle Scholar
  4. 4.
    Allez, R., Dumaz, L.: From sine kernel to Poisson statistics. Elec. J. Probab. 19, 1–25 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Allez, R., Dumaz, L.: Tracy-Widom at high temperature. J. Stat. Phys. 156(6), 1146–1183 (2014)MATHMathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Studies in advanced mathematics. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  7. 7.
    Bao, Z., Su, Z.: Local Semicircle law and Gaussian fluctuation for Hermite \(\beta \)-ensemble. arXiv:1104.3431
  8. 8.
    Ben Arous, G., Guionnet, A.: Large deviations for Wigner’s law and Voiculescu’s non commutative entropy. Probab. Theory Relat. F. 108, 517–542 (1997)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bourgade, P.: Bulk universality for one-dimensional log-gases. In: Proceedings of the XVIIth International Congress On Mathematical Physics, World Scientific Publishing (2013)Google Scholar
  10. 10.
    Bourgade, P., Erdös, L., Yau, H.T.: Bulk universality of general \(\beta \)-ensembles with non-convex potential. J. Math. Phys. 53, 095221 (2012). special issue in honor of E. Lieb’s 80th birthdayMathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Boucheron, S., Lugosi, G., Massart, P.: Concentration Inequalities. Oxford University Press, Oxford (2013)MATHCrossRefGoogle Scholar
  12. 12.
    Bourgade, P., Erdös, L., Yau, H.T.: Edge Universality of \(\beta \)-ensembles. Commun. Math. Phys. (2013)Google Scholar
  13. 13.
    Bourgade, P., Erdös, L., Yau, H.T.: Universality of general \(\beta \)-ensembles. Duke Math. J. 163(6), 1127–1190 (2014)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Durrett, R.: Probability : Theory and Examples, 4th edn. Cambridge University, Cambridge (2010)CrossRefGoogle Scholar
  15. 15.
    Dumitriu, I., Edelman, A.: Matrix models for beta ensembles. J. Math. Phys. 43, 5830–5847 (2002)MATHMathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Enriquez, N., Ménard, L.: Asymptotic expansion of the expected spectral measure of Wigner matrices. arXiv:1506.03002
  17. 17.
    Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91(1), 151–204 (1998)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Killip, R., Stoiciu, M.: Eigenvalue statistics for CMV matrices: from Poisson to clock via random matrix ensembles. Duke Math. J. 146, 361–399 (2009)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Leblé, T., Serfaty, S.: Large deviation principle for empirical fields of Log and Riesz gases. arXiv:1502.02970 (2015)
  20. 20.
    Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, 335. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  21. 21.
    Ramírez, J.A., Rider, B., Virág, B.: Beta ensembles, stochastic Airy spectrum, and a diffusion. J. Am. Math. Soc. 24, 919–944 (2011)MATHCrossRefGoogle Scholar
  22. 22.
    Simon, B.: Trace Ideals and Their Applications. Mathematical Surveys and Monographs 120, 2nd edn. AMS, Providence (2005)MATHGoogle Scholar
  23. 23.
    Sosoe, P., Wong, P.: Local semicircle law in the bulk for Gaussian \(\beta \)-ensemble. J. Stat. Phys. 148(2), 204–232 (2012)MATHMathSciNetCrossRefADSGoogle Scholar
  24. 24.
    Sosoe, P., Wong, P.: Convergence of the eigenvalue density for \(\beta \)-Laguerre ensembles on short scales. Electron. J. Probab. 19(34), 1–18 (2014)MathSciNetGoogle Scholar
  25. 25.
    Trinh, K.D., Tomoyuki, S.: The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles. arXiv:1504.06904
  26. 26.
    Valkó, B., Virág, B.: Continuum limits of random matrices and the Brownian carousel. Invent. Math. 117, 463–508 (2009)CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.MAP 5, UMR CNRS 8145 - Université Paris DescartesParis Cedex 6France
  2. 2.LPMA, Université Paris DiderotParisFrance

Personalised recommendations