Negative-weight Percolation: Review of Existing Literature and Scaling Behavior of the Path Weight in 2d

Abstract

We consider the negative weight percolation (NWP) problem on hypercubic lattice graphs with fully periodic boundary condi- tions in all relevant dimensions from d = 2 to the upper critical dimension d = 6. The problem exhibits edge weights drawn from disorder distributions that allow for weights of either sign. We are interested in the statistical properties of the full ensemble of loops with negative weight, i.e. non-trivial (system spanning) loops as well as topologically trivial (“small”) loops that comprise the “loops only” variant of the NWP problem. The NWP phenomenon refers to the disorder driven proliferation of system span- ning loops of total negative weight. For the numerical simulations we employ a mapping of the NWP model to a combinatorial optimization problem that can be solved exactly by using sophisticated matching algorithms. This allows for the numerically exact study of large systems with good statistics, important to ensure a reliable disorder average. Early simulations for the 2 d setup led to suggest that the resulting negative-weight percolation (NWP) problem is fundamentally different from conventional percolation. Here, we review several studies that reported on results of numerical simulations aimed at clarifying the geometric properties of NWP on hypercubic lattice graphs and random graphs. Finally we present additional new results for the scaling behavior of the geometric properties and the configurational weight of minimum-weight paths (MWPs) in the “loops + MWP” variant of the model, characterizing an additional threshold , above which the disorder averaged MWP weight ( ω p ) is negative, thereby highlighting a characteristic limiting case of the NWP model at small densities of negative edges.