On the transience of random interlacements

TL;DR

This paper proves that the random interlacement graph on Z^d is almost surely transient for any positive level u, advancing understanding of the graph's long-term behavior in probabilistic combinatorics.

Contribution

It establishes the transience of the random interlacement graph for all positive levels u, a key property previously unconfirmed.

Findings

The interlacement graph is almost surely transient for all u>0.

The result applies to the entire class of random interlacement models.

This enhances understanding of the long-term properties of random interlacements.

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 graph induced by the random interlacements at level u of Sznitman arXiv:0704.2560. We prove that for any u>0, almost surely, the random interlacement graph is transient.