Abstract

We consider the interlacement Poisson point process on the space of doubly-infinite Zd-valued trajectories, d >=3. This random spatial process was recently introduced by Sznitman in order to describe the local picture left by the trace of a random walk when it visits a positive fraction of a large d-dimensional torus.

The present talk summarizes recent joint work with Artem Sapozhnikov (ETH). We show that almost surely every two points of the random interlacement are connected via at most ceiling(d/2) trajectories, and that this number is optimal. With a variant of this connectivity argument we also prove that the graph induced by the random interlacements is almost surely transient and that Bernoulli percolation on this graph has a non-trivial phase transition in wide enough slabs.

These results strongly suggest that despite the long-range dependencies present in the model, the geometry of the random interlacement graph is similar to that of the underlying lattice Zd