Deviations from the Circular Law

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Consider Ginibre's ensemble of N × N non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance 1/N. 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 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.