Statistical and dynamical properties of the discrete Sinai model at finite times

We study the Sinai model for the diffusion of a particle in a one-dimensional quenched random energy landscape. We consider the particular case of discrete energy landscapes made of random jumps on the semi-infinite line with a reflecting wall at the origin. We compare the statistical distribution of the successive local minima of the energy landscapes, which we derive explicitly, with the dynamical distribution of the position of the diffusing particle, which we obtain numerically. At high temperature, the two distributions match only in the large time asymptotic regime. At low temperature however, we find even at finite times a clear correspondence between the statistical and dynamical distributions, with additional interesting oscillatory behaviours.