Spectral form factor in a random matrix theory

Abstract

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 $τ<τ_c$ and constant for $τ>τ_c$. Near zero and near $τ_c$ its exhibits oscillations which have been discussed in several recent papers. In the problems of mesoscopic fluctuations and quantum chaos a comparison is often made with random matrix theory. It turns out that, even in the simplest Gaussian unitary ensemble, these oscilllations have not yet been studied there. For random matrices, the two-level correlation function $ρ(λ_1,λ_2)$ 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 $ρ(λ_1,λ_2)$. However the cross-over near zero and $τ_c$ requires 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 cross-over oscillatory properties. The method is also extended to the time-dependent case.