The 2d-directed spanning forest converges to the Brownian web

Abstract

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 $\mathcal{N}$ on ${\mathbb{R}}^{2}$. If the DSF has direction $-e_{y}$, the ancestor $h({\mathbf{u}})$ of a vertex ${\mathbf{u}}\in {\mathcal{N}}$ 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). This verifies a conjecture made by Baccelli and Bordenave in 2007 (Ann. Appl. Probab. 17 (2007) 305–359).