Fluctuation results for size of the vacant set for random walks on discrete torus

We consider one or more independent random walks on the d ≥ 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. Interestingly, unlike the random interlacement model, the fluctuation order has no phase transition. The proof relies on a refined analysis of tail estimates for hitting time and can be applied to other vertex-transitive graphs.