DIRECTED WAVES IN RANDOM-MEDIA - AN ANALYTICAL CALCULATION

The propagation of directed scalar waves in D + 1 dimensions in a strongly disordered medium is studied. We use the model first proposed by Saul, Kardar, and Read [Phys. Rev. A 45, 8859 (1992)], where unitarity is guaranteed in each step. The beam positions [x2BAR] and [x2BAR] characterize the transverse fluctuations of a directed wave front, where the overbar means an average over the wave profile for a given realization of randomness, and [] means a quenched average over all realizations. We introduce G(q)k(y) as the Laplace-transformed Green function of two free random walkers with center-of-mass momentum k and relative position y. We calculate analytically the mean-square deviation of the beam center, [x2BAR], as a function of time. The results show that, for large t, [x2BAR] behaves as (1/square-root pi)t1/2- 1/4 + O(t-3/2) in 1 + 1 dimensions and as (lnt + 4 ln2 + gamma)/4pi + O(1/t) in 2+1 dimensions and takes the finite value 1/2D[G(q=1)k=0 (0) - square-root (27/4pi)t-1/2delta(D,3)] + O(1/t) in D + 1 dimensions where D greater-than-or-equal-to 3, gamma being the Euler constant. We generalize these results to a twofold random walk with any probability-flux-conserving interaction. In all cases the leading term at large t depends solely on the finite value or leading singularity of G(q)k=0(0) at q = 1.