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Local Central Limit Theorem for Determinantal Point Processes

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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.

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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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Melbourne, Melbourne, VIC, 3010, Australia

    Peter J. Forrester

  2. Departments of Mathematics and Physics, Rutgers University, Piscataway Township, NJ, 08854-8019, USA

    Joel L. Lebowitz

  3. Institute for Advanced Study, Princeton, NJ, 08540, USA

    Joel L. Lebowitz

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  1. Peter J. Forrester
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Correspondence to Peter J. Forrester.

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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

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  • DOI: https://doi.org/10.1007/s10955-014-1071-2

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