On the fragmentation of a torus by random walk

We consider a simple random walk on a discrete torus (Z/NZ)^d with dimension d at least 3 and large side length N. For a fixed constant u > 0, we study the percolative properties of the vacant set, consisting of the set of vertices not visited by the random walk in its first [uN^d] steps. We prove the existence of two distinct phases of the vacant set in the following sense: if u > 0 is chosen large enough, all components of the vacant set contain no more than a power of log(N) vertices with high probability as N tends to infinity. On the other hand, for small u > 0, there exists a macroscopic component of the vacant set occupying a non degenerate fraction of the total volume N^d. In dimensions d at least 5, we additionally prove that this macroscopic component is unique, by showing that all other components have volumes of order at most a power of log(N). Our results thus solve open problems posed by Benjamini and Sznitman in arXiv:math/0610802, who studied the small u regime in high dimension. The proofs are based on a coupling of the random walk with random interlacements on Z^d. Among other techniques, the construction of this coupling employs a refined use of discrete potential theory. By itself, this coupling strengthens a result in arXiv:0802.3654.