Dynamical aspects of the quantum chaos border in localisation-delocalisation transitions

Localisation phenomena in quantum systems can occur in very different circumstances: localisation may be induced by disorder, by the integrability or quasi-integrability of an underlying classical dynamics and can be found in non- or weakly interacting multi-particle systems. Such systems can often be described in terms of a Hamiltonian H = Ho + r V, with Ho being diagonal with Poisson distributed entries and V denotes residual interactions. In many physical situations, V is a sparsely filled matrix. We investigate the influence of the structure of the perturbation on the localisation/delocalisation transition as the interaction strength r increases. Using quantum graph techniques we find universal scaling laws as well as dynamical effects based on how fast perturbations can propagate through the system.