Geometry of the random interlacement

TL;DR

This paper investigates the geometric structure of random interlacements on high-dimensional lattices, establishing sharp bounds on the number of trajectories needed to connect points within the interlacement set.

Contribution

It proves that two points in the interlacement set can be connected by at most eil(d/2) trajectories, and shows this bound is optimal.

Findings

Paths between points can be constructed using at most eil(d/2) trajectories.

The bound eil(d/2) is sharp; fewer trajectories may not suffice.

The results utilize stochastic dimension theory to analyze the geometry.

Abstract

We consider the geometry of random interlacements on the $d$-dimensional lattice. We use ideas from stochastic dimension theory developed in \cite{benjamini2004geometry} to prove the following: Given that two vertices $x,y$ belong to the interlacement set, it is possible to find a path between $x$ and $y$ contained in the trace left by at most $\lceil d/2 \rceil$ trajectories from the underlying Poisson point process. Moreover, this result is sharp in the sense that there are pairs of points in the interlacement set which cannot be connected by a path using the traces of at most $\lceil d/2 \rceil-1$ trajectories.