Spectral form factor in a random matrix theory

In the theory of disordered systems the spectral form factor S(τ), the Fourier transform of the two-level correlation function with respect to the difference of energies, is linear for τ<${\mathrm{\tau }}_{\mathrm{c}}$ and constant for τ>${\mathrm{\tau }}_{\mathrm{c}}$. Near zero and near ${\mathrm{\tau }}_{\mathrm{c}}$ it exhibits oscillations which have been discussed in several recent papers. In problems of mesoscopic fluctuations and quantum chaos a comparison is often made with a random matrix theory. It turns out that, even in the simplest Gaussian unitary ensemble, these oscillations have not yet been studied there. For random matrices, the two-level correlation function ρ(${\lambda }_1$,${\lambda }_2$) exhibits several well-known universal properties in the large-N limit. Its Fourier transform is linear as a consequence of the short-distance universality of ρ(${\lambda }_1$,${\lambda }_2$). However the crossover near zero and ${\mathrm{\tau }}_{\mathrm{c}}$ requires one to study these correlations for finite N. For this purpose we use an exact contour-integral representation of the two-level correlation function which allows us to characterize these crossover oscillatory properties. This representation is then extended to the case in which the Hamiltonian is the sum of a deterministic part ${\mathrm{H}}_0$ and of a Gaussian random potential V. Finally, we consider the extension to the time-dependent case.