# Title: Fracturing Ranked Surfaces

**Hans Herrmann**

**Short Abstract**

A “ranked surface” is a lattice on which every site has a rank. Examples are discretized landscapes or sequential percolation. If one cuts the most important connecting bonds a crack appears which has in two dimensions a fractal dimension of 1.217. A classical example is the watershed that separates hydrological basins. In percolation, “bridges” are those sites or bonds which, if occupied, would create the spanning cluster. Suppressing systematically the occupation of these bridges delays the percolation threshold and produces at the end a connected line of bridges which corresponds to the watershed of a random landscape. Also optimal path cracks, the shortest path on loop-less percolation, minimal spanning trees, specific min-max paths and multiple invasion percolation clusters belong to the same universality class. At the percolation threshold bridge percolation exhibits a different exponent, namely ¾, and one finds theta point scaling with a novel crossover exponent. For all dimensions below the upper critical dimension d_c = 6 these exponents are calculated. In dimensions larger than two another universality class appears corresponding to the cutting bonds in percolation, i.e. those bonds which if removed would disconnect the spanning cluster.

### SPONSORS

#### University of Michigan Sponsors

- Horace H. Rackham School of Graduate Studies at the University of Michigan, Ann Arbor
- Michigan Center for Theoretical Physics
- Center for Studies of Complex Systems
- Medical School
- Department of Mathematics
- Department of Physics
- Biophysics Program