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 distributions of the type P(log(10) E)=const that lead to the strong disorder limit. If we truncate the distribution, so that P(log(10) E)=const only for E(min) < or =E < or =E(max), and P(log(10) E)=0 otherwise, we obtain a crossover from self-similar (strong disorder) to self-affine (moderate disorder) behavior at a path length l(x). We find that l(x) proportional, variant[log(10)(E(max)/E(min))](kappa), where the exponent kappa has the value kappa=1.60 +/- 0.03 both in d=2 and d=3. We show how the crossover can be understood from the distribution of local energies on the optimal paths.