F. Seno et al., OPTIMAL SELF-AVOIDING PATHS IN DILUTE RANDOM MEDIUM, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 55(4), 1997, pp. 3859-3864
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 ac
cessible bonds between the undirected and directed percolation thresho
lds, overhangs do not seem to change the standard self-affine scaling
regime characteristic of directed paths. At the undirected threshold t
he path becomes fractal, with a fractal dimension equal to that of the
minimal length path on the infinite cluster backbone. At this point t
he 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 deter
mined by fluctuations in backbone geometry and not by disorder in bond
energies. This scenario is qualitatively confirmed and extended by re
normalization-group calculations on hierarchical lattices.