We consider the random interlacements process with intensity $u$ on ${mathbb Z}^d$, $dge 5$ (call it $I^u$), built from a Poisson point process on the space of doubly infinite nearest neighbor trajectories on ${mathbb Z}^d$. For $kge 3$ we want to de
termine 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 $x_1,...,x_kin mathcal I^u$, 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.
At each point of a Poisson point process of intensity $lambda$ in the hyperbolic place, center a ball of bounded random radius. Consider the probability $P_r$ that from a fixed point, there is some direction in which one can reach distance $r$ withou
t hitting any ball. It is known cite{BJST} that if $lambda$ is strictly smaller than a critical intensity $lambda_{gv}$ then $P_r$ does not go to $0$ as $rto infty$. The main result in this note shows that in the case $lambda=lambda_{gv}$, the probability of reaching distance larger than $r$ decays essentially polynomial, while if $lambda>lambda_{gv}$, the decay is exponential. We also extend these results to various related models.
