Directed negative-weight percolation.

Abstract

We consider the negative-weight percolation model on directed graphs. In particular, we study the model on a two-dimensional, weighted, periodic, centered square lattice. Bond weights are taken from a distribution that allows for both positive and negative values. For a given realization of the disorder, a minimally weighted directed loop and path configuration is determined by performing a nontrivial transformation of the original lattice into a minimum-weight perfect matching problem. For this problem, fast polynomial-time algorithms are available, thus we could study large systems with high accuracy. Depending on the fraction of negatively and positively weighted bonds on the lattice, a continuous phase transition is identified and its characterizing critical exponents are estimated by means of a finite-size scaling analysis. We observe a strong change of the universality class with respect to standard directed percolation as well as with respect to undirected negative-weight percolation. Furthermore, the relation to directed polymers in random media is illustrated.