  We investigate the relation between the local picture left by the trajectory
of a simple random walk on the torus (Z/NZ)^d, d >= 3, until u N^d time steps,
u > 0, and the model of random interlacements recently introduced by Sznitman.
In particular, we show that for large N, the joint distribution of the local
pictures in the neighborhoods of finitely many distant points left by the walk
up to time u N^d converges to independent copies of the random interlacement at
level u.
