RANDOM MATRICES: UNIVERSALITY OF LOCAL SPECTRAL STATISTICS OF NON-HERMITIAN MATRICES

摘要

It is a classical result of Ginibre that the normalized bulk k-point correlation functions of a complex n x n Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on C with kernel K-infinity (z, w):=1/pi e-vertical bar z vertical bar(2)/2-vertical bar w vertical bar(2)/2+z (w) over bar in the limit n ->infinity. In this paper, we show that this asymptotic law is universal among all random n x n matrices M-n whose entries are jointly independent, exponentially decaying, have independent real and imaginary parts and whose moments match that of the complex Gaussian ensemble to fourth order. Analogous results at the edge of the spectrum are also obtained. As an application, we extend a central limit theorem for the number of eigenvalues of complex Gaussian matrices in a small disk to these more general ensembles.