Consider Ginibre's ensemble of N×N non-Hermitian random matrices in
which all entries are independent complex Gaussians of mean zero and variance
N1. As N↑∞ the normalized counting measure of the
eigenvalues converges to the uniform measure on the unit disk in the complex
plane. In this note we describe fluctuations about this {\em Circular Law}.
First we obtain finite N formulas for the covariance of certain linear
statistics of the eigenvalues. Asymptotics of these objects coupled with a
theorem of Costin and Lebowitz then result in central limit theorems for a
variety of these statistics