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

Understand

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 $h(u)$ of a vertex $u \in \mathcal{N}$ 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).