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 [Formula Presented] that lead to the strong disorder limit. If we truncate the distribution, so that [Formula Presented] only for [Formula Presented] and [Formula Presented] otherwise, we obtain a crossover from self-similar (strong disorder) to self-affine (moderate disorder) behavior at a path length [Formula Presented] We find that [Formula Presented] where the exponent [Formula Presented] has the value [Formula Presented] both in [Formula Presented] and [Formula Presented] We show how the crossover can be understood from the distribution of local energies on the optimal paths.