Connectivity properties of random interlacement and intersection of random walks

Abstract

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.

minor corrections

References in corpus (2)

• On the internal distance in the interlacement set
• The effect of small quenched noise on connectivity properties of random interlacements

Cited by in corpus (16)

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• On the internal distance in the interlacement set
• On scaling limits and Brownian interlacements
• The effect of small quenched noise on connectivity properties of random interlacements
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• On the easiest way to connect $k$ points in the Random Interlacements process
• Percolative properties of Brownian interlacements and its vacant set
• Percolation for the Finitary Random interlacements
• Connectivity properties of Branching Interlacements
• The Component Graph of the Uniform Spanning Forest: Transitions in Dimensions $9,10,11,\dots$
• Connectedness of Poisson cylinders in Euclidean space
• Capacity of the range of random walk on ${\mathbb{Z}}^d$
• On the transience of random interlacements
• Random Interlacements via Kuznetsov Measures
• On the exact orders of critical value in Finitary Random Interlacements
• Two observations on the capacity of the range of simple random walks on ${\mathbb{Z}}^3$ and ${\mathbb{Z}}^4$