Abstract
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.
Similar content being viewed by others
References
Akemann, G., Ipsen, J., Kieburg, M.: Products of rectangular random matrices: singular values and progressive scattering. Phys. Rev. E 88, 052118 (2013)
Akemann, G., Kieburg, M., Wei, L.: Singular value correlation functions for products of Wishart matrices. J. Phys. A 46, 275205 (2013)
Bender, E.A.: Central and local limit theorems applied to asymptotic enumeration. J. Combin. Theory Ser. A 15, 91–111 (1973)
Bornemann, F.: On the numerical evaluation of distributions in random matrix theory: a review. Markov Process. Relat. Fields 16, 803–866 (2010)
Bourgade, P., Erdös, L., Yau, H-T.: Edge universality of beta ensembles (2013). arXiv:1306.5728
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)
Costin, O., Lebowitz, J.L.: Gaussian fluctuations in random matrices. Phys. Rev. Lett. 75, 69–72 (1995)
Dobrushin, R.L., Tirozzi, B.: The central limit theorem and the problem of equivalence of ensembles. Commun. Math. Phys. 54, 173–192 (1977)
Dyson, F.J.: Statistical theory of energy levels of complex systems III. J. Math. Phys. 3, 166–175 (1962)
Dyson, F.J., Mehta, M.L.: Statistical theory of energy levels of complex systems IV. J. Math. Phys. 3, 701–712 (1963)
Feller, W.: An Introduction to Probability Theory and its Applications, 2nd edn. Wiley, New York (1971)
Forrester, P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton, NJ (2010)
Forrester, P.J.: Asymptotics of spacing distributions 50 years later (2012). arXiv:1204.3225
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)
Forrester, P.J., Witte, N.S.: Painlevé II in random matrix theory and related fields (2012). arXiv:1210.3381.
Fuji, A.: On the zeros of Dirichlet \(L\)-functions I. Trans. Am. Math. Soc. 196, 225–235 (1974)
Gustavsson, J.: Gaussian fluctuations in the GUE. Ann. l’Inst. Henri Poincaré (B) 41, 151–178 (2005)
Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes. American Mathematical Society, Providence, RI (2009)
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)
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.
Kargin, V.: On Pfaffian random point fields. J. Stat. Phys. 154, 681–704 (2014)
Keating, J.P., Snaith, N.C.: Random matrix theory and \(\zeta (1/2 + it)\). Commun. Math. Phys. 214, 57–89 (2001)
Killip, R.: Gaussian fluctuations for \(\beta \) ensembles. Int. Math. Res. Not. 2008, rnn007 (2008)
Lebowitz, J.L.: Charge fluctuations in Coulomb systems. Phys. Rev. A 27, 1491–1494 (1983)
Lebowitz, J.L., Pittel, B., Ruelle, D., Speer, E.: in preparation.
Maples, K., Rodgers, B.: Bootstrapped zero density estimates and a central limit theorem for the zeros of the zeta function. arXiv:1404.3080.
Martin, PhA, Yalçin, T.: The charge fluctuations in classical Coulomb systems. J. Stat. Phys. 22, 435 (1980)
Mehta, M.L.: Random Matrices, 2nd edn. Academic Press, New York (1991)
Mehta, M.L.: Power series for the level spacing functions of random matrix ensembles. Z. Phys. B 86, 285–290 (1992)
Mehta, M.L., Dyson, F.J.: Statistical theory of the energy levels of complex systems. V. J. Math. Phys. 4, 713–719 (1963)
Niculescu, C.P.: A new look at Newton’s inequalities. J. Inequal. Pure Appl. Math. 1, 17 (2000)
O’Rourke, S.: Gaussian fluctuations of eigenvalues in Wigner random matrices. J. Stat. Phys. 138, 1045–1066 (2010)
Pastur, L., Shcherbina, M.: Eigenvalue Distribution of Large Random Matrices. American Mathematical Society, Providence, RI (2011)
Rodgers, B.: A central limit theorem for the zeros of the zeta function. J. Number Theory 10, 483–511 (2014)
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)
Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55, 923–975 (2000)
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)
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)
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 2nd edn. Cambridge University Press, Cambridge (1965)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Forrester, P.J., Lebowitz, J.L. Local Central Limit Theorem for Determinantal Point Processes. J Stat Phys 157, 60–69 (2014). https://doi.org/10.1007/s10955-014-1071-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-014-1071-2