Gaussian fluctuation for the number of particles in Airy, Bessel, sine, and other determinantal random point fields. (English) Zbl 1041.82001

Soshnikov, Alexander B.

J. Stat. Phys. 100, No. 3-4, 491-522 (2000).

Summary: We prove the central limit theorem (CLT) for the number of eigenvalues near the spectrum edge for certain Hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the Gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions. As another corollary of the Costin-Lebowitz theorem we prove the CLT for the empirical distribution function of the eigenvalues of random matrices from classical compact groups.

MSC:

82B05	Classical equilibrium statistical mechanics (general)
60F99	Limit theorems in probability theory
82B31	Stochastic methods applied to problems in equilibrium statistical mechanics
15B52	Random matrices (algebraic aspects)

Keywords:

determinantal random point fields; central limit theorem; random matrices; Airy kernel; Bessel kernel; classical compact groups