# Seminar at SMU 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.