The noise in the circular law and the Gaussian free field

Fill an n × n matrix with independent complex Gaussians of variance 1/n. As n → ∞, the eigenvalues {zk} converge to a sum of an H1-noise on the unit disk and an independent H1/2-noise on the unit circle. More precisely, for C1 functions of suitable growth, the distribution of ∑n k=1(f(zk) − Ef(zk)) converges to that of a mean-zero Gaussian with variance given by the sum of the squares of the disk H1 and the circle H1/2 norms of f . As a consequence, with pn the characteristic polynomial, it is found that log |pn| −E log |pn| tends to the planar Gaussian free field conditioned to be harmonic outside the unit disk. Further, for polynomial test functions f , we prove that the limiting covariance structure is universal for a class of models including Haar distributed unitary matrices.