  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.
