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Theme for TIFR Centre For Applicable Mathematics, Bangalore

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.