Copyright © The Author 2007. Published by Oxford University Press.
The Noise in the Circular Law and the Gaussian Free Field
Department of Mathematics, University of Colorado at Boulder, UCB 395, Boulder, Colorado 80309
Correspondence: Correspondence to be sent to: Brian Rider, Department of Mathematics, University of Colorado at Boulder, UCB 395, Boulder, Colorado 80309. e-mail: brider{at}euclid.colorado.edu
Fill an n x n matrix with independent complex Gaussians of variance 1/n. As n 
, the eigenvalues {zk} onverge 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 
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.