  We consider one or more independent random walks on the $d\ge 3$ dimensional
discrete torus. The walks start from vertices chosen independently and
uniformly at random. We analyze the fluctuation behavior of the size of some
random sets arising from the trajectories of the random walks at a time
proportional to the size of the torus. Examples include vacant sets and the
intersection of ranges. The proof relies on a refined analysis of tail
estimates for hitting time and can be applied for other vertex-transitive
graphs.
