Optimal Paths and Universality

Optimal paths in disordered systems are studied using two different models interpolating between weak and infinitely strong disorder. In one case, exact numerical methods are used to study the optimal path in a two-dimensional square lattice whereas a renormalization-group analysis is employed on hierarchical lattices in the other. The scaling behaviour is monitored as a function of parameters that tune the strength of the disorder. Two distinct scenarios are provided by the models: in the first, fractal behaviour occurs abruptly as soon as the disorder widens, while in the other it emerges as a limiting case of a self-affine regime.