River basin geomorphology is a very old subject of study initiated by Horton (1945). Various statistical models of drainage networks have been proposed; each such model being a random directed graph with its own nuances. In recent years physicists have been interested in these models because of the commonality of such branching networks in many areas of statistical physics (see Rodriguez-Iturbe and rinaldo (1997) for a detailed survey). We discuss the geometric features of one such model and also study its scaling limit. The scaling limit of this model is the Brownian web, which has lately been the focus of extensive study by probabilists. Using this scaling limit we formalize Hack’s law, which is a proportionality relation between the length and the watershed area of a river.
Biography
Rahul Roy is Professor at Theoretical Statistics and Mathematics Unit of Indian Statistical Institute. His research interests are Probability, Particle systems, Percolation Theory, Stochastic geometry, Random Matrices, History of mathematics