The Brownian vacancy driven walk

We investigate the lattice walk performed by a tagged member of an infinite "sea" of particles filling a d-dimensional lattice, in the presence of a Brownian vacancy. Particle-particle exchange is forbidden; the only interaction between them being hard core exclusion. The tagged particle, differing from the others only by its tag, moves only when it exchanges places with the hole. In this sense, it is a lattice walk "driven" by the Brownian vacancy. The probability distributions for its displacement and for the number of steps taken, after n-steps of the vacancy, are derived. Surprisingly, none of them is a Gaussian! It is shown that the only nontrivial dimensional where the walk is recurrent is d=2.