The Spectral Form Factor Is Not Self-Averaging

The form factor, k (t), is the spectral statistic which best displays nonuniversal quasiclassical deviations from random matrix theory. Recent estimations of k (t) for a single spectrum found interesting new effects of this type. It was supposed that k (t) is self-averaging and thus did not require an ensemble average. We here argue that this supposition sometimes fails and that for many important systems an ensemble average is essential to see detailed properties of k (t). In other systems, notably the nontrivial zeros of Riemann zeta function, it will be possible to see the nonuniversal properties by an analysis of a single spectrum.