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(log₁₀E)=const that lead to the strong disorder limit. If we truncate the distribution, so that P(log₁₀E)=const only for Eₘᵢₙ<~E<~Eₘₐₓ, and P(log₁₀E)=0 otherwise, we obtain a crossover from self-similar (strong disorder) to self-affine (moderate disorder) behavior at a path length l_×. We find that l_×∝[log₁₀(Eₘₐₓ/Eₘᵢₙ)]^κ, where the exponent κ has the value κ=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.