Connectivity properties of random interlacement and intersection of random walks

By Bal'azs R'ath, Artëm Sapozhnikov. 2010

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We consider the interlacement Poisson point process on the space of doubly-infinite Z d -valued trajectories modulo time shift, tending to infinity at pos- itive 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 (2010). 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 dd/2e 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 dd/2e trajectories of the point process. In par- ticular, this implies the already known result of Sznitman (2010) that the random interlacement at level u is connected.

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