Geometry of optimal path hierarchies

We investigate the hierarchy of optimal paths in a disordered landscape, based on the best path, the second best path and so on in terms of an energy. By plotting each path at a height according to its energy above some zero level, a landscape appears. This landscape is self-affine and controlled by two Hurst exponents: the one controlling the height fluctuations is 1/3 and the one controlling the fluctuations of the equipotential lines in the landscape is 2/3. These two exponents correspond to the exponents controlling energy and shape fluctations in the directed polymer problem. We furthermore find that the density of spanning optimal paths scale as the length of the paths to −2/3 and the histogram of energy differences between consecutive paths scale as a power law in the difference size with exponent −2.5.