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

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 R 2 . If the DSF has direction −e y , the ancestor h(u) of a vertex u ∈ 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). See Theorem 1.2. This verifies a conjecture made by Baccelli and Bordenave in 2007 [5].MSC: 60D05.

Transmission and navigation on disordered lattice networks, directed spanning forests and Brownian web

“…Given F n we have the information that the history region H n can not have a perturbed point in its interior and the explored region H n clearly depends on the past movements. At this place it is good to mention here that, as observed for the discrete DSF (defined on a random subset of Z d and studied in [RSS15]), or for the Poisson DSF (studied in [BB07], [CSST19]), it can be shown that the process {(g n (u 1 ),…”

“…In [BB07] it has been shown that scaled paths of the successive ancestors in the DSF converges weakly to the Brownian motion and also conjectured that the scaling limit of the DSF on the Poisson points is the Brownian web. This conjecture remained open for long time and very recently it has been proved in [CSST19]. The scaling limit of the collection of all paths is much harder question as one has to deal with the dependencies between different paths also.…”

IDLA aggregates with infinitely many sources and the directed IDLA forest

“…We are now prepared to prove (12). Indeed, since |i |≤M α N i = 0 if and only if A * n [M α ] = , we have…”

“…The above result is also one of the main ingredients to derive Lemma 4.4. The next one comes from [12] and provides finiteness (and also tail decay but we omit this part here) for the hitting time to 0 for a discrete-time, non-negative valued process {Y t : t ≥ 0} which is not necessarily Markov. Only supermartingale structure and moment conditions for increments are assumed.…”

“…See also [11] in which the author exploits this link between trees and the associated forests to quantify the density of infinite branches of these trees. Furthermore, the associated forest generally is much more than a tool: in [12], for example, the authors prove that the Directed spanning Forest mentioned above converges in distribution to the Brownian web.…”