Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions.

Area of Science:

• Statistical Physics
• Computational Physics
• Materials Science

Background:

• Percolation theory studies the formation of connected clusters in random systems.
• Accurate estimation of critical exponents is crucial for understanding phase transitions.
• Previous methods for constructing large percolation clusters often require significant computational memory.

Purpose of the Study:

• To develop a memory-efficient method for constructing large percolation clusters.
• To accurately estimate the minimum path exponent, d(min), in 4D and 5D hypercubic lattices.
• To precisely determine percolation thresholds (p(c)) and Fisher exponents (tau) for site and bond percolation in these dimensions.

Main Methods:

• A novel cluster construction method limiting memory usage to the size of the largest chemical shell.
• Analysis of both site and bond percolation on 4D and 5D hypercubic lattices.
• High-precision calculation of percolation thresholds and minimum path exponents.

Main Results:

• The memory requirement scales as s^theta, with theta ranging from 0.4 (2D) to 0.5 (>=6D).
• Estimated d(min) values: 1.607+/-0.005 (4D) and 1.812+/-0.006 (5D).
• Precise percolation thresholds and Fisher exponents were determined for 4D and 5D lattices.

Conclusions:

• The developed method enables the creation of very large percolation clusters with minimal memory.
• The study provides highly accurate values for critical exponents in 4D and 5D, refining theoretical models.
• This work offers a robust computational approach for studying complex systems in higher dimensions.