  We study sums of directed paths on a hierarchical lattice where each bond has
either a positive or negative sign with a probability $p$. Such path sums $J$
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 $<J^n>$ always determine,
uniquely, the probability distribution $P(J)$. We also derive, exactly, the
moment behavior as a function of $p$ in the thermodynamic limit. Extrapolations
($n\to 0$) to obtain $<\ln J>$ 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 $p$. Finally, we derive the exact probability distribution for path sums $J$
up to length L=64 for all sign probabilities.
