Directed negative-weight percolation

Abstract

We consider a directed variant of the negative-weight percolation model in a two-dimensional, periodic, square lattice. The problem exhibits edge weights which are taken from a distribution that allows for both positive and negative values. Additionally, in this model variant all edges are directed. For a given realization of the disorder, a minimally weighted loop/path configuration is determined by performing a non-trivial 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 edges in the lattice, a continuous phase transition can be identified, whose characterizing critical exponents we have estimated by a finite-size scaling analyses of the numerically obtained data. 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.

From Spin Glasses to Negative-Weight Percolation

“…Recently, NWP was studied on directed graphs [ 91 ]. The algorithmic procedure for directed graphs is explained in Section 4.5 .…”

“…Therefore, before running proper simulations, the ratio has to be determined by first ( Figure 21 , right) [ 93 ]. A detailed analysis [ 91 ] exhibits that two-dimensional directed NWP is in a universality class different from undirected NWP and also different from directed standard percolation.…”