Random Walks, Reaction-Diffusion, and Nonequilibrium Dynamics of Spin Chains in One-Dimensional Random Environments

Sinai's model of diffusion in one dimension with random local bias is studied by a real space renormalization group which yields asymptotically exact long time results. The distribution of the position of a particle and the probability of it not returning to the origin are obtained, as well as the two-time distribution which exhibits ``aging'' with lnt/lnt^' scaling and a singularity at x(t)0ex0ex=0ex0exx(t^'). The effects of a small uniform force are also studied. Extension to motion of many domain walls yields nonequilibrium time dependent correlations for the 1D random field Ising model with Glauber dynamics and ``persistence'' exponents of 1D reaction-diffusion models with random forces.