Spectral form factor in a random matrix theory

In the theory of disordered systems the spectral form factor S(tau), the Fourier transform of the two-level correlation function with respect to the difference of energies, is linear for tau<tau(c) and constant for tau>tau(c). Near zero and near tau(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 rho(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 rho(lambda(1),lambda(2)) However the crossover near zero and tau(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 H-0 and of a Gaussian random potential V. Finally, we consider the extension to the time-dependent case.