Tracer particle motion in a two-dimensional lattice gas with low vacancy density

We investigate the random motion of tracer particles in a two-dimensional lattice gas with a low density ϱ of vacancies. It is shown by direct calculation that, on a square lattice, a tracer particle's mean square displacement 〈y2〉t increases as ϱt(π − 1) for all times t å 1, the unit of time being the inverse step frequency of a single vacancy. However, for the distribution function of the displacement, Pt(y), two time regimes have to distinguished: an initial regime t å ϱ −1ln ϱ −1, where it is a modified Bessel function, and a regime t å ϱ −1ln ϱ −1 where it becomes Gaussian. We calculate the scaling function which describes the crossover between the two regimes in the limit of low vacancy density, long times, and large distances. The analogous problem is considered on a strip of finite width L, where the crossover takes place for t ∼ ϱ −2L−2.