  We consider the set of points visited by the random walk on the discrete
torus $(\mathbb{Z}/N\mathbb{Z})^d$, for $d \geq 3$, at times of order $uN^d$,
for a parameter $u>0$ in the large-$N$ limit. We prove that the vacant set left
by the walk undergoes a phase transition across a non-degenerate critical value
$u_* = u_*(d)$, as follows. For all $u< u_*$, the vacant set contains a giant
connected component with high probability, which has a non-vanishing asymptotic
density and satisfies a certain local uniqueness property. In stark contrast,
for all $u> u_*$ the vacant set scatters into tiny connected components. Our
results further imply that the threshold $u_*$ precisely equals the critical
value, introduced by Sznitman in arXiv:0704.2560, which characterizes the
percolation transition of the corresponding local limit, the vacant set of
random interlacements on $\mathbb{Z}^d$. Our findings also yield the analogous
infinite-volume result, i.e. the long purported equality of three critical
parameters $\bar u$, $u_*$ and $u_{**}$ naturally associated to the vacant set
of random interlacements.
