  Fill an n x n matrix with independent complex Gaussians of variance 1/n. As n
approaches infinity, the eigenvalues {z_k} converge to a sum of an H^1-noise on
the unit disk and an independent H^{1/2}-noise on the unit circle. More
precisely, for C^1 functions of suitable growth, the distribution of
sum_{k=1}^n (f(z_k)-E f(z_k)) converges to that of a mean-zero Gaussian with
variance given by the sum of the squares of the disk H^1 and the circle H^{1/2}
norms of f. Moreover, with p_n the characteristic polynomial, log|p_n|- E
log|p_n| tends to the planar Gaussian free field conditioned to be harmonic
outside the unit disk. Finally, for polynomial test functions f, we prove that
the limiting covariance structure is universal for a class of models including
Haar distributed unitary matrices.
