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We consider a simple random walk on 
$\mathbb{Z}^d$ started at the origin and stopped on its first exit time from 
$({-}L,L)^d \cap \mathbb{Z}^d$. Write L in the form 
$L = m N$ with 
$m = m(N)$ and N an integer going to infinity in such a way that 
$L^2 \sim A N^d$ for some real constant 
$A \gt 0$. Our main result is that for 
$d \ge 3$, the projection of the stopped trajectory to the N-torus locally converges, away from the origin, to an interlacement process at level 
$A d \sigma_1$, where 
$\sigma_1$ 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).
We study the geometry of the component of the origin in the uniform spanning forest of 
$\mathbb{Z}^{d}$ and give bounds on the size of balls in the intrinsic metric.
