We consider the geometry of random interlacements on the $d$-dimensional lattice. We use ideas from stochastic dimension theory developed in  to prove the following: Given that two vertices $x,y$ belong to the interlacement set, it is possible to find a path between $x$ and $y$ contained in the trace left by at most $\lceil d/2 \rceil$ trajectories from the underlying Poisson point process. Moreover, this result is sharp in the sense that there are pairs of points in the interlacement set which cannot be connected by a path using the traces of at most $\lceil d/2 \rceil-1$ trajectories.
"Geometry of the random interlacement." Electron. Commun. Probab. 16 528 - 544, 2011. https://doi.org/10.1214/ECP.v16-1660