Journal of Statistical Physics

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

Local Central Limit Theorem for Determinantal Point Processes

Article

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.

Keywords

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

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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

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