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June 2016 Hack’s law in a drainage network model: A Brownian web approach
Rahul Roy, Kumarjit Saha, Anish Sarkar
Ann. Appl. Probab. 26(3): 1807-1836 (June 2016). DOI: 10.1214/15-AAP1134


Hack [Studies of longitudinal stream profiles in Virginia and Maryland (1957). Report], 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. We study the tributary structure of Howard’s drainage network model of headward growth and branching studied by Gangopadhyay, Roy and Sarkar [Ann. Appl. Probab. 14 (2004) 1242–1266]. We show that the exponent of Hack’s law is $2/3$ for Howard’s model. Our study is based on a scaling of the process whereby the limit of the watershed area of a stream is area of a Brownian excursion process. 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.


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Rahul Roy. Kumarjit Saha. Anish Sarkar. "Hack’s law in a drainage network model: A Brownian web approach." Ann. Appl. Probab. 26 (3) 1807 - 1836, June 2016.


Received: 1 February 2015; Revised: 1 July 2015; Published: June 2016
First available in Project Euclid: 14 June 2016

zbMATH: 1344.60017
MathSciNet: MR3513607
Digital Object Identifier: 10.1214/15-AAP1134

Primary: 60D05

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.26 • No. 3 • June 2016
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