Averaged Greenʹs functions of discrete disordered systems from effective energy-dependent probability distributions

In a discrete disordered system one is interested in computing the averaged Greenʹs function Gij . Using the supersymmetry formulation as a starting point, we derive a renormalization group flow equation for the effective probability distribution of a subsystem of fixed size, which preserves Gij , as the size of the total system is increased. From this flow equation, averaged Greenʹs functions can be computed directly in the thermodynamic limit, which enables us to compute the density of states and investigate localization/delocalization transitions. As an illustration, we consider the one-dimensional tight-binding Anderson model with Lorentz disorder.