Directed paths on hierarchical lattices with random sign weights

We study sums of directed paths on a hierarchical lattice where each bond has either a positive or negative sign with a probability [formula presented] Such path sums [formula presented] have been used to model interference effects by hopping electrons in the strongly localized regime. The advantage of hierarchical lattices is that they include path crossings, ignored by mean field approaches, while still permitting analytical treatment. Here we perform a scaling analysis of the controversial “sign transition” using Monte Carlo sampling, and conclude that the transition exists and is second order. Furthermore, we make use of exact moment recursion relations to find that the moments [formula presented] always determine, uniquely, the probability distribution [formula presented] We also derive, exactly, the moment behavior as a function of [formula presented] in the thermodynamic limit. Extrapolations [formula presented] to obtain [formula presented] for odd and even moments yield a new signal for the transition that coincides with Monte Carlo simulations. Analysis of high moments yield interesting “solitonic” structures that propagate as a function of [formula presented] Finally, we derive the exact probability distribution for path sums [formula presented] up to length [formula presented] for all sign probabilities.