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On the easiest way to connect $$k$$ points in the random interlacements process. (English) Zbl 1277.60177
Summary: We consider the random interlacements process with intensity $$u$$ on $$\mathbb Z^d, d \geq 5$$ (call it $$\mathcal I^u$$), built from a Poisson point process on the space of doubly infinite nearest neighbor trajectories on $$\mathbb Z^d$$. For $$k \geq 3$$ 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 $$x_1 , \ldots, x_k\in \mathcal I^u$$, there is a sequence of $$n(k, d)$$ trajectories $$\gamma^1 , \ldots, \gamma^{n(k,d)}$$ from the underlying Poisson point process such that the union of their traces $$\bigcup^{n(k,d)}_{i=1}\text{Tr}(\gamma^i)$$ is a connected set containing $$x_1 , \ldots, x_k$$. Moreover we show that this result is sharp, i.e. that a.s. one can find $$x_1 , \ldots, x_k\in \mathcal I^u$$ that cannot be linked together by $$n(k, d) - 1$$ trajectories.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
##### Keywords:
random interlacement; connectivity; percolation; random walk
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