Optimal self-avoiding paths in dilute random medium

The combined effects of bond-energy disorder and random-bond exclusion on optimal undirected self-avoiding paths are studied by an original finite-size scaling method in two dimensions. For concentrations of accessible bonds between the undirected and directed percolation thresholds, overhangs do not seem to change the standard self-affine scaling regime characteristic of directed paths. At the undirected threshold the path becomes fractal, with a fractal dimension equal to that of the minimal length path on the infinite cluster backbone. At this point the optimal energy variance scales with time t as t(omega c) (omega(c)= 1.02+/-0.05). Furthermore, omega(c) turns out to be exclusively determined by fluctuations in backbone geometry and not by disorder in bond energies. This scenario is qualitatively confirmed and extended by renormalization-group calculations on hierarchical lattices.