Fluctuation formula for complex random matrices

A Gaussian fluctuation formula is proved for linear statistics of complex r andom matrices in the case where the statistic is rotationally invariant. F or a general linear statistic without this symmetry, Coulomb gas theory is used to predict that the distribution will again be a Gaussian, with a spec ific mean and variance. The variance splits naturally into a bulk and surfa ce contibution, the latter resulting from the long-range correlations at th e boundary of the support of the eigenvalue density.