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).
Citation
David Coupier. Kumarjit Saha. Anish Sarkar. Viet Chi Tran. "The 2d-directed spanning forest converges to the Brownian web." Ann. Probab. 49 (1) 435 - 484, January 2021. https://doi.org/10.1214/20-AOP1478
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