  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 +/- 1 jumps on the semi infinite line Z+ 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 behaviors.
