Journal of Statistical Physics, Vol. 65, Nos. 1/2, 1991. p. 183-204.
We consider the minimal paths on a hierarchical diamond lattice, where bonds are assigned a random weight. Depending on the initial distribution of weights, we find all possible asymptotic scaling properties. The different cases found are the small-disorder case, the analog of Levy's distributions with a power-law decay at -oo and finally a limit of large disorder which can be identified as a percolation problem. The asymptotic shape of the stable distributions of weights of the minimal path are obtained, as well as their scaling properties. As a side result, we obtMn the asymptotic form of the distribution of effective percolation thresholds for finite-size hierarchical lattices.