For d >= 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 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 star(d) of random interlacements on Z(d). Our main tool is a variant of the soft-local time coupling technique of Popov and Teixeira [J. Eur. Math. Soc. (JEMS) 17 (2015) 2545-2593].
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