Random directed forest and the Brownian web

Abstract

It is shown that this random graph is a tree almost surely for d = 2 and 3 and it is an infinite collection of disjoint trees for d ≥ 4. In addition, for d = 2, we show that when properly scaled, the family of its paths converge in distribution to the Brownian web.

The Brownian Web, the Brownian Net, and their Universality

“…Because the dependence is in some sense local in all the models described above, it is natural to conjecture that for d = 1, the collection of directed paths in the drainage network (after diffusive scaling) should converge to the Brownian web. This has indeed been verified for various drainage networks on Z 1+1 [CDF09, CV11,RSS16a], and for the Poisson trees on R 1+1 [FFW05]. The main difficulty in these studies lies in the dependence among the paths.…”

“…Interestingly, the image of ∂K 0 under F n forms a coalescing flow on the circle, where for x, y ∈ ∂K 0 ⊂ ∂K n , the length of the arc between F n (x) and F n (y) on the unit circle is proportional to the probability that a new particle will be attached to the corresponding part of ∂K n between x and y. In the limit that the particle radius δ ↓ 0, while time is sped up by a factor of δ −3 , this coalescing flow can be seen as a localized disturbance flow studied in [NT15], which converges to the coalescing Brownian flow on the circle Figure 19: From left to right, illustration of paths in the drainage network of [GRS04], the drainage network of [RSS16a], Poisson trees [FLT04], the directed spanning forest [BB07], and the radial spanning tree [BB07]. w.r.t.…”

“…From each source (x, s), a directed edge is then drawn to the closest source in the next layer Z d × {s + 1}, and ties are broken by choosing each closest source with equal probability. There are also other variants such as in [ARS08], where the directed edge from the source (x, s) ∈ Z d+1 connects to the closest source in the 45 o light cone rooted at (x, s), or as in [RSS16a] where the directed edge from (x, s) connects to the closest source in Z d × {s, s + 1, . .…”

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

“…As explained above, this is no longer true for the DSF because of complex geometrical dependencies. Recently, several papers [25,31]-Saha and Sarkar are involved in the first one -have considered modifications of the DSF in order to make the problem more tractable but until this paper, the conjecture of Baccelli and Bordenave remained open.…”

“…We also think that Theorem 5.2 is interesting in itself and very robust. In particular, it should provide the required coalescence time estimates for all the drainage network models in the basin of attraction of the BW [9,11,16,25,31]. See Remark 5.8 for further details.…”