On the easiest way to connect $k$ points in the Random Interlacements process

We consider the random interlacements process with intensity $u$ on ${\mathbb Z}^d$, $d\ge 5$ (call it $I^u$), built from a Poisson point process on the space of doubly infinite nearest neighbor trajectories on ${\mathbb Z}^d$. For $k\ge 3$ we want to determine the minimal number of trajectories from the point process that is needed to link together $k$ points in $\mathcal I^u$. Let $$n(k,d):=\lceil \frac d 2 (k-1) \rceil - (k-2).$$ We prove that almost surely given any $k$ points $x_1,...,x_k\in \mathcal I^u$, there is a sequence ofof $n(k,d)$ trajectories $\gamma^1,...,\gamma^{n(k,d)}$ from the underlying Poisson point process such that the union of their traces $\bigcup_{i=1}^{n(k,d)}\tr(\gamma^{i})$ is a connected set containing $x_1,...,x_k$. Moreover we show that this result is sharp, i.e. that a.s. one can find $x_1,...,x_k in I^u$ that cannot be linked together by $n(k,d)-1$ trajectories.