Theoretical Statistics and Mathematics Unit, ISI Delhi

August 19, 2015 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Kumarjit Saha,
Indian Statistical Institute, Delhi

Title:
Random directed trees and their scaling limit

Abstract of Talk

The study of the similarities between river basins and the reasons behind their existence are the core of hydrology. There are are mainly two independent power laws observed for major river basins. One of them is Hack's law. Hack (1957), while studying the drainage system in the Shenandoah valley and the adjacent mountains of Virginia, observed a power law relation $l \sim a^{0.6}$ between the length $l$ of a stream from its source to a divide and the area $a$ of the basin that collects the precipitation contributing to the stream as tributaries. In this talk we consider the tributary structure of Howard's drainage network model, studied by Gangopadhyay et al. (2004), and show that the exponent of Hack's law is $2/3$ for Howard's model. To obtain this we define a dual of the model and show that under diffusive scaling, both the original network and its dual converge jointly to the standard Brownian web and its dual. For different drainage network models it is natural to ask whether the resulting graph is a single tree or not. We consider a discrete drainage model - discrete directed spanning forest (DDSF) constructed as follows. Each vertex in ${\mathbb Z}^d$ is open or closed with probability $p$ or $(1-p)$ respectively. An open vertex ${\mathbf u} = ({\mathbf u}(1),\ldots,{\mathbf u}(d))$ connected by an edge to another open vertex which has the minimum $L_1$ distance among all the open vertices with ${\mathbf v}(d) > {\mathbf u}(d)$. 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 $d\geq 4$. In addition for $d=2$ we show that when properly scaled, family of its paths converges in distribution to the Brownian web.

These are joint work with Rahul Roy and Anish Sarkar.