title: Upper critical dimension of the negative-weight percolation problem.
abstract: By means of numerical simulations, we investigate the geometric properties of loops on hypercubic lattice graphs in dimensions d=2 through 7, where edge weights are drawn from a distribution that allows for positive and negative weights. We are interested in the appearance of system-spanning loops of total negative weight. The resulting negative-weight percolation (NWP) problem is fundamentally different from conventional percolation, as we have seen in previous studies of this model for the two-dimensional case. Here, we characterize the transition for hypercubic systems, where the aim of the present study is to get a grip on the upper critical dimension d u of the NWP problem. 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. We characterize the loops via observables similar to those in percolation theory and perform finite-size scaling analyses, e.g., three-dimensional hypercubic systems with side length up to L=56 sites, in order to estimate the critical properties of the NWP phenomenon. We find our numerical results consistent with an upper critical dimension d u=6 for the NWP problem.
authors.affiliation: Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany. oliver.melchert@uni-oldenburg.de
authors.forename: O
authors.initials: O
authors.lastname: Melchert
authors.forename: L
authors.initials: L
authors.lastname: Apolo
authors.forename: A K
authors.initials: AK
authors.lastname: Hartmann