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 $P\left({\log }_{10}E\right)=\mathrm{const}$ that lead to the strong disorder limit. If we truncate the distribution, so that $P\left({\log }_{10}E\right)=\mathrm{const}$ only for $E_{\min }<~E<~E_{\max },$ and $P\left({\log }_{10}E\right)=0\mathrm{}$ otherwise, we obtain a crossover from self-similar (strong disorder) to self-affine (moderate disorder) behavior at a path length ${\mathcal{l}}_{\times }.$ We find that ${\mathcal{l}}_{\times }\propto \left[{\log }_{10}{(E}_{\max }{/E}_{\min }\right){]}^{\kappa },$ where the exponent $\kappa$ has the value $\kappa =1.60\mathrm{}\pm 0.03$ both in $d=2\mathrm{}$ and $d=3.\mathrm{}$ We show how the crossover can be understood from the distribution of local energies on the optimal paths.