Percolation thresholds and Fisher exponents in hypercubic lattices

Area of Science:

• Statistical Physics
• Computational Physics

Background:

• Percolation theory studies the connectivity of random networks.
• Understanding critical phenomena in higher dimensions is crucial for various scientific fields.

Purpose of the Study:

• To compute accurate percolation thresholds (p_c) for hypercubic lattices Z^d.
• To determine the Fisher exponent (τ) governing cluster size distribution at criticality.
• To investigate the validity of mean-field theory predictions in higher dimensions.

Main Methods:

• Invasion percolation algorithm was employed for numerical simulations.
• Calculations were performed on hypercubic lattices Z^d for dimensions d=4 to 13.
• Analysis focused on bond and site percolation thresholds and the Fisher exponent.

Main Results:

• Highly accurate numerical values for bond and site percolation thresholds were obtained.
• The Fisher exponent (τ) was computed for dimensions d=4 to 13.
• Results support the mean-field value τ=5/2 for d≥6.
• Logarithmic corrections to power-law scaling were observed at d=6.

Conclusions:

• The study provides precise numerical evidence for percolation phenomena in higher dimensions.
• The findings validate theoretical predictions regarding the Fisher exponent in the critical regime.
• The transition from lower to higher dimensional behavior is elucidated.