Interlacement limit of a stopped random walk trace on a torus

We consider a simple random walk on Z _d started at the origin and stopped on its first exit time from {equation presented}. Write L in the form L = mN with m = m(N) and N an integer going to infinity in such a way that L ² ∼ AN ^d for some real constant A > 0. Our main result is that for d ≥ 3, the projection of the stopped trajectory to the N-torus locally converges, away from the origin, to an interlacement process at level Adσ ₁, where σ ₁ is the exit time of a Brownian motion from the unit cube (-1, 1) ^d that is independent of the interlacement process. The above problem is a variation on results of Windisch (2008) and Sznitman (2009).