Power spectra of Dyson's circular ensembles

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 $k$ eigenvalues in an interval. These authors moreover showed how for the circular unitary ensemble integrability properties of the generating function, via a particular Painlev\'e VI system, imply a computational scheme for the corresponding power spectrum, and allow for the determination of its large $N$ limit. In the present work, these results are extended to the case of the circular orthogonal ensemble and circular symplectic ensemble, where the integrability is expressed through four particular Painlev\'e VI systems for finite $N$, and two Painlev\'e III$'$ systems for the limit $N\to\infty$, and also via corresponding Fredholm determinants. The relation between the limiting power spectrum $S_\infty(\omega)$, where $\omega$ denotes the Fourier variable, and the limiting generating function for the conditioned gap probabilities is particular direct, involving just a single integration over the gap endpoint in the latter. Interpreting this generating function as the characteristic function of a counting statistic allows for it to be shown that $S_\infty(\omega) \mathop{\sim} \limits_{\omega \to 0} {1 \over \pi \beta | \omega|}$, where $\beta$ is the Dyson index.