Optimal paths in disordered media: Scaling of the crossover from self-similar to self-affine behavior

We study optimal paths in disordered energy landscapes using energy distrib utions of the type P(log(10)E) = const that lead to the strong disorder lim it. If we truncate the distribution, so that P(log(10)E)= const only for E- min less than or equal to E less than or equal to E-max, and P(log(10)E) = 0 otherwise, we obtain a crossover from self-similar (strong disorder) to s elf-affine (moderate disorder) behavior at a path length l(x). We find that l(x) proportional to[log(10)(E-max/E-min)](kappa), where the exponent kapp a has the value kappa = 1.60+/-0.03 both in d=2 and d=3. We show how the cr ossover can be understood from the distribution of local energies on the op timal paths. [S1063-651X(99)51409-8].