The 2D-directed spanning forest converges to the Brownian web (Q2227719)

Consider a homogeneous Poisson point process \(\mathcal{N}\) in \(\mathbb{R}^2\) with intensity \(\lambda\). The horizontal and vertical axes are interpreted as space and time axes, respectively. The ancestor of \(\mathbf{x}\in\mathcal{N}\) is the closest (in the Euclidean distance) Poisson point \(\mathbf{y}=h(\mathbf{x})\) whose second coordinate (time) exceeds the second coordinate of \(\mathbf{x}\). The Directed Spanning Forest (DSF) is the random geometric graph with vertex set \(\mathcal{N}\) and edges \((\mathbf{x},h(\mathbf{x}))\). This object was introduced by \textit{F. Baccelli} and \textit{C. Bordenave} [Ann. Appl. Probab. 17, No. 1, 305--359 (2007; Zbl 1136.60007)], who showed that under diffusive scaling the trajectory of DSF converges in distribution to a Brownian motion. The authors prove a stronger result, which establishes a convergence of a properly scaled DSF to the Brownian web, thereby setting a conjecture by Baccelli and Bordenave.