Abstract.
Consider Ginibre’s ensemble of N×N non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance 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.
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Acknowledgments.
A serious debt of gratitude is due to the referee. In addition to helping correct various small glitches, their thoroughness identified some important errors in an earlier version of this paper.
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This work was supported in part by the grant NSF-9983320.
Revised version: 13 February 2004
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Rider, B. Deviations from the Circular Law. Probab. Theory Relat. Fields 130, 337–367 (2004). https://doi.org/10.1007/s00440-004-0355-x
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DOI: https://doi.org/10.1007/s00440-004-0355-x