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

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

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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

Information

Received: 1 June 2018; Revised: 1 September 2020; Published: January 2021
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.1214/20-AOP1478

Subjects:
Primary: 60D05

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 1 • January 2021
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