Exact results and self-averaging properties for random-random walks on a one-dimensional infinite lattice

We present new exact results for a one-dimensional asymmetric disordered hopping model. The lattice is taken infinite from the start and we do not resort to the periodization scheme used by Derrida. An explicit resummation allows for the calculation of the velocityV and the diffusion constantD (which are found to coincide with those given by Derrida) and for demonstrating thatV is indeed a self-averaging quantity; the same property is established forD in the limiting case of a directed walk.