Directed waves in random media

We investigate an alternative model for the propagation of directed waves in strongly disordered media. The basic ansatz of our approach is that impurity scattering events can be described by the action of random S matrices. This approach has two important advantages over those considered in previous works. First, it yields a numerical discretization in which unitarity is manifestly preserved. Second, the model enables one to compute certain averages over disorder exactly. The beam positions [〈x²〉] and [〈x〉²] characterize the transverse fluctuations of a directed wave front, where 〈...〉 indicates an average over the wave profile for a given realization of randomness, and [...] indicates quenched averaging over all realizations. We confirm the well-known result that the beam width [〈x²〉] grows linearly with the propagation distance, t. We also obtain numerically in two and three dimensions (2D and 3D) the behavior of the beam center [〈x〉²] as a function of t. The results suggest that [〈x〉²] scales as t1/2 in 2D and as lnt in 3D. We show how these scaling laws emerge in a natural way from the problem of two interacting random walkers. Connections to the problem of directed polymers in random media are also explored.