Power spectra of Dyson's circular ensembles

Abstract

The power spectrum is a Fourier series statistic associated with the covariances of the displacement from average positions of the members of an eigen-sequence. When this eigen-sequence has rotational invariance, as for the eigen-angles of Dyson's circular ensembles, recent work of Riser and Kanzieper has uncovered an exact identity expressing the power spectrum in terms of the generating function for the conditioned gap probability of having k=0,…,N eigenvalues in an interval. These authors moreover showed how for the circular unitary ensemble integrability properties of the generating function, via a particular Painlevé VI system, imply a computational scheme for the corresponding power sp..