Theoretical Statistics and Mathematics Unit, ISI Delhi

August 27, 2014 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Kumarjit Saha,
Indian Statistical Institute, Delhi

Title:
Some quantitative estimates in a drainage network model

Abstract of Talk

Hack, [1957] while studying the drainage system in the Shenandoah valley
and the adjacent mountains of Virginia
observed a power law relation $L = 1.4 A^{0.6}$
between the length $L$ of a stream from its
source to a divide and the watershed area $A$ formed by its tributaries
upstream from the divide.
In this work we study the tributary structure of
a 2-dimensional drainage network model known as Howard's model of headward growth
and branching studied by Gangopadyay, Roy and Sarkar [2004].
Our study is based on a scaling of the process and
we obtain the watershed area of a stream as the area of a Brownian
excursion process. This gives a statistical
explanation of Hack's law and justifies the remark of Giacometti {\it et al.}
\textit{``From the results we suggest that a statistical framework referring
to the scaling invariance of the entire basin structure should be used in
the interpretation of Hack's law.''}
The proof uses the convergence of drainage network to the Brownian web under
suitable scaling (Coletti, Fontes, Dias [2009]).
The method of the proof is quite general and can be used for other drainage network models also
which under suitable scaling converge to the Brownian web.
This is a joint work with Professor Rahul Roy and Professor Anish Sarkar.