Random walkers in one-dimensional random environments: exact renormalization group analysis.

Sinai's model of diffusion in one dimension with random local bias is studied by a real space renormalization group, which yields exact results at long times. The effects of an additional small uniform bias force are also studied. We obtain analytically the scaling form of the distribution of the position x(t) of a particle, the probability of it not returning to the origin, and the distributions of first passage times, in an infinite sample as well as in the presence of a boundary and in a finite but large sample. We compute the distribution of the meeting time of two particles in the same environment. We also obtain a detailed analytic description of the thermally averaged trajectories by computing quantities such as the joint distribution of the number of returns and of the number of jumps forward. These quantities obey multifractal scaling, characterized by generalized persistence exponents theta(g) which we compute. In the presence of a small bias, the number of returns to the origin becomes finite, characterized by a universal scaling function which we obtain. The full statistics of the distribution of successive times of return of thermally averaged trajectories is obtained, as well as detailed analytical information about correlations between directions and times of successive jumps. The two-time distribution of the positions of a particle, x(t) and x(t') with t>t', is also computed exactly. It is found to exhibit "aging" with several time regimes characterized by different behaviors. In the unbiased case, for t-t' approximately t'alpha with alpha>1, it exhibits a ln t/ln t' scaling, with a singularity at coinciding rescaled positions x(t)=x(t'). This singularity is a novel feature, and corresponds to particles that remain in a renormalized valley. For closer times alpha<1, the two-time diffusion front exhibits a quasiequilibrium regime with a ln(t-t')/ln t' behavior which we compute. The crossover to a t/t' aging form in the presence of a small bias is also obtained analytically. Rare events corresponding to intermittent splitting of the thermal packet between separated wells which dominate some averaged observables are also characterized in detail. Connections with the Green function of a one-dimensional Schrödinger problem and quantum spin chains are discussed.