The noise in the circular law and the Gaussian free field

Details The noise in the circular law and the Gaussian free field The noise in the circular law and the Gaussian free field

2006-06-27

arxiv.org/abs/math/0606663v2

Int. Math. Res. Not. 2007, no. 2, Art. ID rnm006, 33 pp

Mathematics

Probability

30 pages, 5 figures. Revised introduction. New sections

Scientific paper

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.

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