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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.

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)
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