Random oriented trees: A model of drainage networks

Abstract

Consider the d-dimensional lattice Z d where each vertex is "open" or "closed" with probability p or 1 − p, respectively. An open vertex v is connected by an edge to the closest open vertex w such that the dth co-ordinates of v and w satisfy w(d) = v(d) − 1. In case of nonuniqueness of such a vertex w, we choose any one of the closest vertices with equal probability and independently of the other random mechanisms. It is shown that this random graph is a tree almost surely for d = 2 and 3 and it is an infinite collection of distinct trees for d ≥ 4. In addition, for any dimension, we show that there is no bi-infinite path in the tree and we also obtain central limit theorems of (a) the number of vertices of a fixed degree ν and (b) the number of edges of a fixed length l.

Random directed forest and the Brownian web

“…Let H be the space of compact subsets of (Π, d Π ) equipped with the Hausdorff metric d H given by, Ferrari et al [10] have shown that, for d = 2, the random graph on the Poisson points introduced by [12], converges to a Brownian web under a suitable diffusive scaling. Coletti et al [7] have a similar result for the discrete random graph studied in Gangopadhyay et al [13]. Baccelli et al [5] have 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 is the Brownian web.…”

“…The graph studied in [13] connected an open vertex u to the vertex h(u) with h(u) being the nearest open vertex in {w : w(d) = u(d) + 1}, with the vertex being chosen with uniform probability in case there are more than one nearest open vertex. This construction immediately leads to a Markovian analysis which is exploited in [13] to obtain the tree/forest dichotomy. However the DSF model considered here has to take care of a 'history' set arising from the paths constructed in the past.…”

The Brownian Web, the Brownian Net, and their Universality

“…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.…”

Hack’s law in a drainage network model: A Brownian web approach

“…From the construction, it follows that the random graph, G = (V, E) with edge set E := { u, h(u) : u ∈ V }, does not contain any circuit. This model has been studied by Gangopadhyay, Roy and Sarkar [10] and the following results were obtained. (i) For d = 2 and d = 3, G consists of one single tree almost surely, and for d ≥ 4, G is a forest consisting of infinitely many disjoint trees almost surely.…”