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. ✷

The directed last-passage percolation (LPP) model with independent exponential times is considered. We complete the study of asymptotic directions of infinite geodesics, started by Ferrari and Pimentel [5]. In particular, using a recent result of [3] and a local modification argument, we prove there is no (random) direction with more than two geodesics with probability 1.

The two-dimensional directed spanning forest (DSF) introduced by Baccelli and Bordenave is a planar directed forest whose vertex set is given by a homogeneous Poisson point process N on R 2 . If the DSF has direction −e y , the ancestor h(u) of a vertex u ∈ N is the nearest Poisson point (in the L 2 distance) having strictly larger y-coordinate. This construction induces complex geometrical dependencies. In this paper we show that the collection of DSF paths, properly scaled, converges in distribution to the Brownian web (BW). See Theorem 1.2. This verifies a conjecture made by Baccelli and Bordenave in 2007 [5].MSC: 60D05.

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