Connectivity properties of random interlacement and intersection of random walks

We consider the interlacement Poisson point process on the space of doubly-infinite Z^d-valued trajectories modulo time-shift, tending to infinity at positive and negative infinite times. The set of vertices and edges visited by at least one of these trajectories is the random interlacement at level u of Sznitman arXiv:0704.2560 . We prove that for any u>0, almost surely, (1) any two vertices in the random interlacement at level u are connected via at most ceiling(d/2) trajectories of the point process, and (2) there are vertices in the random interlacement at level u which can only be connected via at least ceiling(d/2) trajectories of the point process. In particular, this implies the already known result of Sznitman arXiv:0704.2560 that the random interlacement at level u is connected.