Connectivity properties of random interlacement and intersection of random walks

TL;DR

This paper investigates the connectivity properties of the random interlacement set in high-dimensional lattices, establishing bounds on the number of trajectories needed to connect vertices and confirming the set's overall connectivity.

Contribution

It provides new bounds on the number of trajectories required for connectivity in the random interlacement, extending previous results on its connectedness.

Findings

Any two vertices are connected via at most ceiling(d/2) trajectories.

Some vertices require at least ceiling(d/2) trajectories for connection.

The random interlacement at level u is almost surely connected.

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.