Universal scaling functions for bond percolation on planar-random and square lattices with multiple percolating...

Area of Science:

• Statistical physics
• Complex systems
• Network science

Background:

• Percolation theory is crucial for understanding connectivity in random systems.
• Models with multiple percolating clusters are gaining research interest.
• Lattice properties significantly influence percolation phenomena.

Purpose of the Study:

• To investigate bond percolation on diverse lattice structures with varying conditions.
• To analyze the behavior of multiple percolating clusters.
• To identify universal scaling functions in percolation models.

Main Methods:

• Monte Carlo simulations were employed.
• Studied bond percolation on planar random lattices, duals of random lattices, and square lattices.
• Varied boundary conditions (free and periodic) and aspect ratios (L1/L2).

Main Results:

• Calculated probabilities for n percolating clusters (W(n)) and percolating probabilities (P).
• Determined average fractions of lattice bonds/sites in percolating clusters ((n), (n)).
• Established universal finite-size scaling functions for W(n), P, (n), (n), and related probability distributions across lattice types.

Conclusions:

• Universal finite-size scaling functions govern percolation behavior across different lattices.
• Nonuniversal metric factors are consistent across various lattice structures.
• These factors are independent of boundary conditions and aspect ratios, highlighting robustness.