Watershed Percolation
Hans Herrmann
ETH Zurich
Lundi 02/02/2015, 14:00-15:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
In percolation, ``bridges'' are those sites or bonds which if occupied would create the spanning cluster. Suppressing systematically the occupation of bridges delays the percolation threshold and produces at the end a connected line of bridges which corresponds to the watershed line on a random landscape and exhibits trivially a first order transition point. Several other realizations of first order percolation transitions have been proposed recently, like the largest cluster model and the Gaussian model which yield peaked cluster size distributions and compact clusters. The surface of these clusters is fractal and also has the same dimension as the watershed. In fact also optimal path cracks and the shortest path on loop-less or multiple invasion percolation clusters belong also to the same universality class. At $p_c$ bridge percolation exhibits theta point scaling with a novel tricritical exponent for which values in all dimensions below the upper critical dimension $d_c = 6$ are calculated. For weak disorder optimal path cracks have dangling ends and isolated clusters also yielding novel exponents. The watershed also follows SLE properties and describes the crack of the random fuse model in the limit of infinite disorder. The scaling of the bridge bonds can be used to prove the transition between first and second order in a model that avoids spanning.
Contact : Marco SCHIRO


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