Distribution of the Oscillation Period in the Underdamped One Dimensional Sinai Model

We consider the Newtonian dynamics of a massive particle in a one dimemsional random potential which is a Brownian motion in space. This is the zero temperature nondamped Sinai model. As there is no dissipation the particle oscillates between two turning points where its kinetic energy becomes zero. The period of oscillation is a random variable fluctuating from sample to sample of the random potential. We compute the probability distribution of this period exactly and show that it has a power law tail for large period, P(T)\sim T^{-5/3} and an essential singluarity P(T)\sim \exp(-1/T) as T\to 0. Our exact results are confirmed by numerical simulations and also via a simple scaling argument.