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

2018/05/23 by Coupier, David, Saha, Kumarjit, Sarkar, Anish +1 · 2 citations
#60D05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1805.09399

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 N on ℝ². If the DSF has direction -e_y, 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.

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• The Directed Spanning Forest: coalescence versus dimension

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