Continuum random tree as a scaling limit for drainage network models

Abstract: People are interested about finding scaling limits of discrete random trees conditioned to be large. Most notably in the case of a Galton-Watson tree with a finite variance critical offspring distribution and conditioned to have a large number of vertices, Aldous [93] proved that the scaling limit is a continuous random tree called the Brownian CRT. 

For Scheidegger model of river network, the cluster at the origin gives a rooted finite tree with root at the origin and each edge having weight 1 (which is essentially the time distance covered by each edge). Aldous [93] conjectured that such a tree (conditioned to be large) with scaling factor $1/n$ should scale to a continuum random tree like object. We prove that Aldous conjecture is true with a slightly modified scaling limit than what Aldous initially guessed. Our result is a universality class type of result in the sense that the same limit should hold for other drainage network models also in the basin of attraction of the Brownian web.