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arXiv · 1010.0773

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

David Coupier·Viet Chi Tran
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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.

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David Coupier, Viet Chi Tran. 2010-10-05. The 2d-Directed Spanning Forest is almost surely a tree. https://doi.org/10.1002/rsa.20400

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