The upper critical dimension of the negative-weight percolation problem

TL;DR

This study uses numerical simulations to analyze the geometric properties of negative-weight loops on hypercubic lattices in dimensions 2 to 7, identifying the upper critical dimension as 6.

Contribution

It determines the upper critical dimension of the negative-weight percolation problem through finite-size scaling analysis in higher dimensions.

Findings

Upper critical dimension d_u=6 for NWP.

Finite-size scaling confirms critical behavior in dimensions up to 7.

Mapping NWP to a combinatorial optimization problem enables exact solutions.

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 2d 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. 3d 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.