Self-averaging characteristics of spectral fluctuations

The spectral form factor as well as the two-point correlator of the density of (quasi-)energy levels of individual quantum dynamics are not self-averaging. Only suitable smoothing turns them into useful characteristics of spectra. We present numerical data for a fully chaotic kicked top, employing two types of smoothing: one involves primitives of the spectral correlator, the second, a small imaginary part of the quasi-energy. Self-averaging universal (like the circular unitary ensemble (CUE) average) behavior is found for the smoothed correlator, apart from noise which shrinks like $1/\sqrt{N}$ as the dimension N of the quantum Hilbert space grows. There are periodically repeated quasi-energy windows of correlation decay and revival wherein the smoothed correlation remains finite as $N\to \infty$ such that the noise is negligible. In between those windows (where the CUE averaged correlator takes on values of the order $1/N^2$) the noise becomes dominant and self-averaging is lost. We conclude that the noise forbids distinction of CUE and GUE-type behavior. Surprisingly, the underlying smoothed generating function does not enjoy any self-averaging outside the range of its variables relevant for determining the two-point correlator (and certain higher-order ones). We corroborate our numerical findings for the noise by analytically determining the CUE variance of the smoothed single-matrix correlator. (paper)