Fluctuations and ergodicity of the form factor of quantum propagators and random unitary matrices

We consider the spectral form factor of random unitary matrices as well as of Floquet matrices of kicked tops, as given by the (squared moduli of) the traces t(n) = tr F-n With the integer 'time' n = 0, +/-1, +/-2,.... For a typical matrix F the rime dependence of the form factor \t(n)\(2) looks err atic; only after a local time average over a suitably large time window Del ta n does a systematic time dependence become manifest. For matrices drawn from the circular unitary ensemble we prove egodicity: in the limits of lar ge matrix dimension and time window Delta n the local time average has vani shingly small ensemble fluctuations and may be identified with the ensemble average. By numerically diagonalizing Floquet matrices of kicked tops with a globally chaotic classical limit we find the same ergodicity. As a bypro duct we find that the traces t(n) of random matrices from the circular ense mbles behave very much like independent Gaussian random numbers. Again, Flo quet matrices of chaotic tops share that universal behaviour. It becomes cl ear that the form factor of chaotic dynamical systems can be fully faithful to random-matrix theory, not only in its locally time-averaged systematic time dependence but also in its fluctuations.