By: Nuno Araújo
From: ETH Zurich
At: Instituto de Investigação Interdisciplinar, Anfiteatro
Discretized landscapes can be mapped onto ranked surfaces, where every element (site or bond) has a unique rank associated with its corresponding relative height . When the landscape is flooded according to this rank and global connectivity is suppressed, a fractal line emerges, which is the watershed of the original landscape. This line hides a new tricritical point at a critical fraction of flooded elements and, in the continuum limit, it is a Schramm-Loewner evolution (SLE) curve . A general description in the context of ranked surfaces allows unveiling how several seemingly unrelated physical models tumble into the same universality class [1, 3]. The generalization to ranked volumes (in three dimensions) provides the framework to determine the effective shares when different companies or nations extract either oil, gas, or water, from the same porous soil .
 K. J. Schrenk, N. A. M. AraÃºjo, J. S. Andrade Jr., and H. J. Herrmann. Fracturing ranked surfaces. Sci. Rep. 2, 348 (2012).
 E. Daryaei, N. A. M. AraÃºjo, K. J. Schrenk, S. Rouhani, and H. J. Herrmann. Watersheds are Schramm-Loewner evolution curves. Phys. Rev. Lett. 109, 218701 (2012).
 A. A. Moreira, C. L. N. Oliveira, A. Hansen, N. A. M. AraÃºjo, H. J. Herrmann, and J. S. Andrade Jr. Fracturing highly disordered materials. Phys. Rev. Lett. 109, 255701 (2012).
 K. J. Schrenk, N. A. M. AraÃºjo, and H. J. Herrmann. How to share underground reservoirs. Sci. Rep. 2, 751 (2012).