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Drainage networks and the Brownian web

Hold Date 2016-09-29 10:30~2016-09-29 15:00

Place Seminar Room W1-D-725, West Zone 1, Ito campus, Kyushu University

Object person  

Speaker Rahul Roy (Indian Statistical Institute)

Time: 10:30 -12:00, 13:30 - 15:00

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 is a random directed graph with its own nuances. The scaling limit of these models is the Brownian web, which has lately been the focus of extensive study among probabilists.

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 Rodrguez-Iturbe and Rinaldo (1997) for a detailed survey).

In these lectures, we first present the geometric properties of the discrete random graphs which are the drainage networks. We then introduce the Brownian web and then show how the scaling limit of these discrete random graphs is the Brownian web. We also establish mathematically an empirical law, called Hack's law, in the study of river basins.

The lectures will be based on some recent works, viz. Roy, Saha and Sarkar (2016, Ann. Inst. H. Poincare) and Roy, Saha and Sarkar (2016, Ann. Appl. Probab.).