Abstract

The random transfer matrices of a quasi-1D system naturally act on the Grassmannian of symplectic frames, which is a compact symmetric space. The random phase hypothesis (RPH) claims that the invariant distribution is given by the Haar measure on the elliptic (or open) channels. This allows to derive the DKMP equations which in turn have a large number of applications. A general criterion is presented allowing to verify a weak form of the RPH in a perturbative situation. The criterion can be verified for the Wegner N-orbital model and this allows to prove that its Lyapunov spectrum is equidistant. Other applications concern the localization length of the Anderson model on a strip (giving a generalization of the 1d-Thouless formula) as well as a perturbative formula for the density of states.