We prove a local central limit theorem (LCLT) for the number of points N(J) in a region J in (Formula presented.) specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of N(J) tends to infinity as (Formula presented.). 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 (Formula presented.)—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.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
- Central limit theorem
- Lee–Yang zeros
- Local central limit theorem
- Random matrices