  For $d\ge 3$ we construct a new coupling of the trace left by a random walk
on a large $d$-dimensional discrete torus with the random interlacements on
$\mathbb Z^d$. This coupling has the advantage of working up to macroscopic
subsets of the torus. As an application, we show a sharp phase transition for
the diameter of the component of the vacant set on the torus containing a given
point. The threshold where this phase transition takes place coincides with the
critical value $u_*(d)$ of random interlacements on $\mathbb Z^d$. Our main
tool is a variant of the soft-local time coupling technique of [PT12].
