The 2d-Directed Spanning Forest is almost surely a tree

TL;DR

This paper proves that the 2D-Directed Spanning Forest constructed from a Poisson point process is almost surely a tree, with no bi-infinite paths, using percolation theory and comparison arguments.

Contribution

It establishes that the DSF is a tree and extends the result to cases with points removed via an auxiliary Boolean model.

Findings

The DSF is almost surely a tree.

No bi-infinite paths exist in the DSF.

The result holds even after removing points from an auxiliary Boolean model.

Abstract

We consider the Directed Spanning Forest (DSF) constructed as follows: given a Poisson point process N on the plane, the ancestor of each point is the nearest vertex of N having a strictly larger abscissa. We prove that the DSF is actually a tree. Contrary to other directed forests of the literature, no Markovian process can be introduced to study the paths in our DSF. Our proof is based on a comparison argument between surface and perimeter from percolation theory. We then show that this result still holds when the points of N belonging to an auxiliary Boolean model are removed. Using these results, we prove that there is no bi-infinite paths in the DSF.