Statistical and dynamical properties of the discrete Sinai model at finitetimes

We study the Sinai model for the diffusion of a particle in a one-dimension al quenched random energy landscape. We consider the particular case of dis crete energy landscapes made of random +/-1 jumps on the semi-infinite line z(+) with a reflecting wall at the origin. We compare the statistical dist ribution of the successive local minima of the energy landscapes, which we derive explicitly, with the dynamical distribution of the position of the d iffusing particle, which we obtain numerically. At high temperature, the tw o distributions match only in the large time asymptotic regime. At low temp erature however, we find even at finite times a clear correspondence betwee n the statistical and dynamical distributions; with additional interesting oscillatory behaviours.