Local Central Limit Theorem for Determinantal Point Processes

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 $|J|\to \infty$. This extends a previous result giving a weaker central limit theorem (CLT) 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 (GSE). 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 neigbors in the bulk.

12 pages; claims relating to LCLT for Pfaffian point processes of version 1 withdrawn in version 2 and replaced by determinantal point processes; improved presentation version 3

References in corpus (4)

• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• Edge Universality of Beta Ensembles
• Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory
• Bootstrapped zero density estimates and a central limit theorem for the zeros of the zeta function

Cited by in corpus (5)

• Random matrices and entanglement entropy of trapped Fermi gases
• Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions
• Central limit theorems, Lee-Yang zeros, and graph-counting polynomials
• Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices
• A note on Lee-Yang zeros in the negative half-plane