Analysis of the loop length distribution for the negative-weight percolation problem in dimensions d=2 through d=6

Area of Science:

• Statistical Physics
• Complex Systems
• Network Science

Background:

• The negative weight percolation (NWP) problem investigates systems with edge weights that can be positive or negative.
• Previous research focused on system-spanning loops in NWP, leaving the behavior of smaller loops less understood.
• Understanding loop ensembles is crucial for characterizing phase transitions in disordered systems.

Purpose of the Study:

• To characterize the loop length distribution of small loops in the negative weight percolation problem.
• To determine critical exponents governing small loop behavior at and below the critical point.
• To relate findings on small loops to existing results for system-spanning loops.

Main Methods:

• Numerical simulations on hypercubic lattice graphs with periodic boundary conditions (d=2 to 6).
• Mapping the NWP model to a combinatorial optimization problem solvable by exact matching algorithms.
• Extensive and exhaustive numerical study to achieve high statistics on very large systems.

Main Results:

• Characterization of the loop length distribution for small loops in NWP.
• Determination of two independent critical exponents describing small loop behavior.
• The results provide a link between small loop statistics and system-spanning loop properties.

Conclusions:

• This study provides a comprehensive numerical analysis of small loops in NWP.
• The identified critical exponents offer new insights into the universality classes of NWP.
• The employed exact combinatorial optimization method enables precise large-scale simulations.