Lower Limit of Percolation Threshold on Square Lattice with Complex Neighborhoods

Area of Science:

• Statistical Physics
• Computational Physics

Background:

• Revisiting the 60-year-old concept of long-range interactions in percolation theory.
• Investigating Dalton, Domb, and Sykes's foundational work on percolation problems.

Purpose of the Study:

• To estimate percolation thresholds for random site percolation on a square lattice with extended neighborhoods.
• To analyze the impact of neighborhood size on percolation thresholds and their scaling behavior.

Main Methods:

• Utilizing Monte Carlo simulations based on the Newman-Ziff algorithm.
• Applying the finite-size scaling hypothesis for accurate threshold estimation.
• Estimating 64 distinct percolation thresholds for varying neighborhood sizes.

Main Results:

• Percolation thresholds range from 0.27013 to 0.11535 as neighborhood size increases.
• Observed a power-law dependence of the percolation threshold on the effective coordination number (exponent ≈ -1/2).
• Empirically determined the limit of percolation threshold scaling with the inverse square of the mean neighborhood radius.

Conclusions:

• Long-range interactions significantly influence percolation thresholds.
• The study confirms established scaling laws and provides empirical limits for complex neighborhood percolation.
• Findings are relevant for understanding phase transitions in systems with extended interactions.