A remark on diffusion of directed polymers in random environments

We consider a system of random walks or directed polymers interacting with an environment which is random in space and time. Under minimal assumptions on the distribution of the environment, we prove that this system has diffusive behavior with probability one if d > 2 and beta < beta(0), where beta(0) is defined in terms of the probability that the symmetric nearest neighbor random walk on the d-dimensional integer lattice ever returns to its starting point. We also obtain a precise estimate for the mean square displacement of this system.