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

Sinai's model of diffusion in one dimension with random local bias is studi ed by a real space renormalization group, which yields exact results at lon g limes. The effects of an additional small uniform bias force are also stu died. We obtain analytically the scaling form of the distribution of the po sition x(t) of a particle, the probability of it not returning to the origi n, and the distributions of first passage times, in an infinite sample as w ell as in the presence of a boundary and in a finite but large sample. We c ompute the distribution of the meeting time of two particles in the same en vironment. We also obtain a detailed analytic description of the thermally averaged trajectories by computing quantities such as the joint distributio n of the number of returns and of the number of jumps forward. These quanti ties obey multifractal scaling, characterized by generalized persistence ex ponents theta(g) which we compute. In the presence of a small bias, the num ber of returns to the origin becomes finite, characterized by a universal s caling 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 dire ctions and times of successive jumps. The two-time distribution of the posi tions 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 dif ferent behaviors. In the unbiased case, for t-t'similar to t'(alpha) with a lpha> 1, it exhibits a In t/ln t' scaling, with a singularity at coinciding rescaled positions x(t) =x(t'). This singularity is a novel feature, and c orresponds to particles that remain in a renormalized valley. For closer ti mes alpha<1, the two-time diffusion front exhibits a quasiequilibrium regim e with a In(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. R are events corresponding to intermittent splitting of the thermal packet be tween separated wells which dominate some averaged observables are also cha racterized in detail. Connections with the Green function of a one-dimensio nal Schrodinger problem and quantum spin chains are discussed. [S1063-651X( 99)06204-2].