  Consider Ginibre's ensemble of $N \times N$ non-Hermitian random matrices in
which all entries are independent complex Gaussians of mean zero and variance
$\frac{1}{N}$. As $N \uparrow \infty$ 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.
