Journal of Statistical Physics

, Volume 157, Issue 1, pp 60–69 | Cite as

Local Central Limit Theorem for Determinantal Point Processes

  • Peter J. ForresterEmail author
  • Joel L. Lebowitz


We prove a local central limit theorem (LCLT) for the number of points \(N(J)\) in a region \(J\) in \(\mathbb R^d\) specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of \(N(J)\) tends to infinity as \(|J| \rightarrow \infty \). This extends a previous result giving a weaker central limit theorem for these systems. Our result relies on the fact that the Lee–Yang zeros of the generating function for \(\{E(k;J)\}\)—the probabilities of there being exactly \(k\) points in \(J\)—all lie on the negative real \(z\)-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble. A LCLT is also established for the probability density function of the \(k\)-th largest eigenvalue at the soft edge, and of the spacing between \(k\)-th neighbors in the bulk.


Central limit theorem Local central limit theorem Lee–Yang zeros Random matrices 



The work of PJF was supported by the Australian Research Council. The work of JLL was supported by NSF Grant DMR1104500. JLL thanks B. Pittel, D. Ruelle and particularly E. Speer for very helpful information about LCLT. We thank T. Spencer and H.-T. Yau for the invitation to participate in the IAS Princeton program ‘Non-equilibrium dynamics and random matrices’, thus facilitating the present collaboration, and we thank H. Spohn and P. Sarnak for comments on various drafts.


  1. 1.
    Akemann, G., Ipsen, J., Kieburg, M.: Products of rectangular random matrices: singular values and progressive scattering. Phys. Rev. E 88, 052118 (2013)CrossRefADSGoogle Scholar
  2. 2.
    Akemann, G., Kieburg, M., Wei, L.: Singular value correlation functions for products of Wishart matrices. J. Phys. A 46, 275205 (2013)CrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Bender, E.A.: Central and local limit theorems applied to asymptotic enumeration. J. Combin. Theory Ser. A 15, 91–111 (1973)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bornemann, F.: On the numerical evaluation of distributions in random matrix theory: a review. Markov Process. Relat. Fields 16, 803–866 (2010)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Bourgade, P., Erdös, L., Yau, H-T.: Edge universality of beta ensembles (2013). arXiv:1306.5728
  6. 6.
    Brody, T.A., Flores, J., French, J.B., Mello, P.A., Pandey, A., Wong, S.S.M.: Random matrix theory. Rev. Mod. Phys. 53, 329–351 (1981)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Costin, O., Lebowitz, J.L.: Gaussian fluctuations in random matrices. Phys. Rev. Lett. 75, 69–72 (1995)CrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Dobrushin, R.L., Tirozzi, B.: The central limit theorem and the problem of equivalence of ensembles. Commun. Math. Phys. 54, 173–192 (1977)CrossRefzbMATHMathSciNetADSGoogle Scholar
  9. 9.
    Dyson, F.J.: Statistical theory of energy levels of complex systems III. J. Math. Phys. 3, 166–175 (1962)CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Dyson, F.J., Mehta, M.L.: Statistical theory of energy levels of complex systems IV. J. Math. Phys. 3, 701–712 (1963)CrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Feller, W.: An Introduction to Probability Theory and its Applications, 2nd edn. Wiley, New York (1971)zbMATHGoogle Scholar
  12. 12.
    Forrester, P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton, NJ (2010)zbMATHGoogle Scholar
  13. 13.
    Forrester, P.J.: Asymptotics of spacing distributions 50 years later (2012). arXiv:1204.3225
  14. 14.
    Forrester, P.J., Rains, E.M.: Inter-relationships between orthogonal, unitary and symplectic matrix ensembles. In: Bleher, P.M., Its, A.R. (eds.), Random matrix models and their applications, Mathematical Sciences Research Institute Publications, vol. 40, pp. 171–208. Cambridge University Press, Cambridge (2001)Google Scholar
  15. 15.
    Forrester, P.J., Witte, N.S.: Painlevé II in random matrix theory and related fields (2012). arXiv:1210.3381.
  16. 16.
    Fuji, A.: On the zeros of Dirichlet \(L\)-functions I. Trans. Am. Math. Soc. 196, 225–235 (1974)Google Scholar
  17. 17.
    Gustavsson, J.: Gaussian fluctuations in the GUE. Ann. l’Inst. Henri Poincaré (B) 41, 151–178 (2005)CrossRefzbMATHMathSciNetADSGoogle Scholar
  18. 18.
    Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes. American Mathematical Society, Providence, RI (2009)zbMATHGoogle Scholar
  19. 19.
    Its, A.R., Kuilaars, A.B.J., Östensson, J.: Critical edge behaviour in unitary random matrix ensembles and the thirty-fourth Painlevé transcendent. IMRN 2008, rnn017 (2008)Google Scholar
  20. 20.
    Kujlaars, A.B.J., Zhang, L.: Singular values of products of Gaussian random ma- trices, multiple orthogonal polynomials and hard edge scaling limits. arXiv:1308.1003.
  21. 21.
    Kargin, V.: On Pfaffian random point fields. J. Stat. Phys. 154, 681–704 (2014)Google Scholar
  22. 22.
    Keating, J.P., Snaith, N.C.: Random matrix theory and \(\zeta (1/2 + it)\). Commun. Math. Phys. 214, 57–89 (2001)CrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Killip, R.: Gaussian fluctuations for \(\beta \) ensembles. Int. Math. Res. Not. 2008, rnn007 (2008)MathSciNetGoogle Scholar
  24. 24.
    Lebowitz, J.L.: Charge fluctuations in Coulomb systems. Phys. Rev. A 27, 1491–1494 (1983)CrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Lebowitz, J.L., Pittel, B., Ruelle, D., Speer, E.: in preparation.Google Scholar
  26. 26.
    Maples, K., Rodgers, B.: Bootstrapped zero density estimates and a central limit theorem for the zeros of the zeta function. arXiv:1404.3080.
  27. 27.
    Martin, PhA, Yalçin, T.: The charge fluctuations in classical Coulomb systems. J. Stat. Phys. 22, 435 (1980)CrossRefADSGoogle Scholar
  28. 28.
    Mehta, M.L.: Random Matrices, 2nd edn. Academic Press, New York (1991)zbMATHGoogle Scholar
  29. 29.
    Mehta, M.L.: Power series for the level spacing functions of random matrix ensembles. Z. Phys. B 86, 285–290 (1992)CrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Mehta, M.L., Dyson, F.J.: Statistical theory of the energy levels of complex systems. V. J. Math. Phys. 4, 713–719 (1963)CrossRefzbMATHMathSciNetADSGoogle Scholar
  31. 31.
    Niculescu, C.P.: A new look at Newton’s inequalities. J. Inequal. Pure Appl. Math. 1, 17 (2000)MathSciNetGoogle Scholar
  32. 32.
    O’Rourke, S.: Gaussian fluctuations of eigenvalues in Wigner random matrices. J. Stat. Phys. 138, 1045–1066 (2010)CrossRefzbMATHMathSciNetADSGoogle Scholar
  33. 33.
    Pastur, L., Shcherbina, M.: Eigenvalue Distribution of Large Random Matrices. American Mathematical Society, Providence, RI (2011)CrossRefzbMATHGoogle Scholar
  34. 34.
    Rodgers, B.: A central limit theorem for the zeros of the zeta function. J. Number Theory 10, 483–511 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants. I Fermion, Poisson and boson point processes. J. Funct. Anal. 205, 414–463 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55, 923–975 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Soshnikov, A.B.: Gaussian fluctuation for the number of particles in Airy, Bessel, Sine, and other determinantal random point fields. J. Stat. Phys. 100, 491–522 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Torquato, S., Scardicchio, A., Zachary, C.E.: Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory. J. Stat. Mech. 2008, P110019 (2008)CrossRefGoogle Scholar
  39. 39.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 2nd edn. Cambridge University Press, Cambridge (1965)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia
  2. 2.Departments of Mathematics and PhysicsRutgers UniversityPiscataway TownshipUSA
  3. 3.Institute for Advanced StudyPrincetonUSA

Personalised recommendations