Directed paths on hierarchical lattices with random sign weights

TL;DR

This paper investigates the behavior of sums of directed paths with random signs on hierarchical lattices, revealing a second-order sign transition, exact distribution formulas, and novel solitonic structures through analytical and Monte Carlo methods.

Contribution

It provides the first detailed analysis of the sign transition on hierarchical lattices, including exact moment relations, probability distributions, and the discovery of propagating solitonic structures.

Findings

Confirmed the existence of a second-order sign transition.

Derived exact probability distributions for path sums.

Identified solitonic structures in high moments as a function of p.

Abstract

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.