We consider a simple random walk on started at the origin and stopped on its first exit time from . Write L in the form with and N an integer going to infinity in such a way that for some real constant . Our main result is that for , the projection of the stopped trajectory to the N-torus locally converges, away from the origin, to an interlacement process at level , where is the exit time of a Brownian motion from the unit cube that is independent of the interlacement process. The above problem is a variation on results of Windisch (2008) and Sznitman (2009).