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

TL;DR

This paper proves that the scaled two-dimensional directed spanning forest (DSF) converges to the Brownian web, confirming a conjecture and revealing the asymptotic behavior of complex planar directed forests.

Contribution

It establishes the convergence of the DSF to the Brownian web, providing a rigorous link between a Poisson-based forest model and continuous stochastic processes.

Findings

DSF paths, when scaled, converge to the Brownian web.

Confirms the 2007 conjecture by Baccelli and Bordenave.

Provides insights into the geometric dependencies of the DSF.

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 $h(u)$ 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). This verifies a conjecture made by Baccelli and Bordenave in 2007.