Geometry of the random interlacement

We consider the geometry of random interlacements on the d-dimensional lattice. We use ideas from stochastic dimension theory developed in [BKPS04] 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 d/2e 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 [d/2]−1 trajectories.