The 2d-directed spanning forest converges to the Brownian web
The Annals of Probability
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 R2. If the DSF has direction −ey, the ancestor h(u) of a vertex u ∈ N is the nearest Poisson point (in the L2 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)
Coupier, David; Saha, Kumarjit; Sarkar, Anish; and Tran, Viet Chi, "The 2d-directed spanning forest converges to the Brownian web" (2021). ISI Best Publications. 26.