Geometry of the random interlacement

Artikel i vetenskaplig tidskrift, 2011

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 ⌈d/2⌉ 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.