We perform numerical simulations to study the optimal path problem on
disordered hierarchical graphs with effective dimension d=2.32. Therein, edge
energies are drawn from a disorder distribution that allows for positive and
negative energies. This induces a behavior which is fundamentally different
from the case where all energies are positive, only. Upon changing the
subtleties of the distribution, the scaling of the minimum energy path length
exhibits a transition from self-affine to self-similar. We analyze the precise
scaling of the path length and the associated ground-state energy fluctuations
in the vincinity of the disorder critical point, using a decimation procedure
for huge graphs. Further, using an importance sampling procedure in the
disorder we compute the negative-energy tails of the ground-state energy
distribution up to 12 standard deviations away from its mean. We find that the
asymptotic behavior of the negative-energy tail is in agreement with a
Tracy-Widom distribution. Further, the characteristic scaling of the tail can
be related to the ground-state energy flucutations, similar as for the directed
polymer in a random medium.Comment: 10 pages, 10 figures, 3 table
