Upper critical dimension of the negative-weight percolation problem

Area of Science:

• Statistical Physics
• Complex Systems
• Computational Physics

Background:

• Negative-weight percolation (NWP) presents unique challenges compared to conventional percolation.
• Understanding NWP's behavior across different dimensions is crucial for complex systems analysis.

Purpose of the Study:

• To investigate the geometric properties of loops in hypercubic lattice graphs with positive and negative edge weights.
• To determine the upper critical dimension (du) for the negative-weight percolation problem.

Main Methods:

• Numerical simulations on hypercubic lattices (d=2 to 7) with a distribution of positive and negative edge weights.
• Mapping the NWP model to a combinatorial optimization problem solvable via matching algorithms.
• Finite-size scaling analyses using observables analogous to percolation theory.

Main Results:

• System-spanning loops of negative total weight are investigated.
• The transition behavior of NWP in hypercubic systems is characterized.
• Numerical results indicate an upper critical dimension (du) of 6 for the NWP problem.

Conclusions:

• The upper critical dimension for negative-weight percolation on hypercubic lattices is found to be 6.
• This finding highlights the distinct nature of NWP compared to traditional percolation theory.
• The study provides a robust characterization of NWP phenomena in higher dimensions.