Finite-size scaling in stick percolation

Area of Science:

• Statistical Physics
• Complex Systems
• Computational Physics

Background:

• Finite-size scaling functions are crucial for understanding phase transitions in physical systems.
• Continuum percolation, particularly with geometric objects like sticks, presents unique challenges compared to lattice models.
• Previous studies have explored lattice percolation, but generalizing to continuum models requires new theoretical and computational approaches.

Purpose of the Study:

• To generalize the concept of universal finite-size scaling functions to continuum percolation models.
• To accurately determine the percolation threshold for isotropic widthless stick systems using advanced simulation techniques.
• To investigate the relationship between stick percolation and lattice percolation by examining their scaling behaviors.

Main Methods:

• Development of a high-efficiency algorithm for Monte Carlo simulations to handle large-size systems.
• Extensive simulations with numerous realizations to ensure statistical robustness.
• Precise determination of the percolation threshold (Nc*l^2) for stick percolation.

Main Results:

• The percolation threshold for isotropic widthless stick systems was determined with high precision: Nc*l^2 = 5.63726 ± 0.00002.
• A nonuniversal metric factor (A = 0.106910 ± 0.000009) was identified.
• The spanning probability of stick percolation, when scaled by this factor, aligns with the universal scaling function of lattice percolation.

Conclusions:

• The study successfully extends finite-size scaling concepts to continuum percolation.
• The findings establish a quantitative link between stick percolation and lattice percolation, demonstrating universality.
• This work provides a robust method for simulating and analyzing continuum percolation phenomena.