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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.

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Artem Sapozhnikov