Percolation thresholds on high-dimensional D_{n} and E_{8}-related lattices

Area of Science:

• Statistical Physics
• Condensed Matter Physics
• Computational Physics

Background:

• Percolation theory is conventionally studied on cubic lattices.
• High-dimensional lattices like D_n and E8 are now accessible via advanced simulation algorithms.
• Understanding percolation on these complex structures is crucial for various scientific fields.

Purpose of the Study:

• To investigate site and bond percolation thresholds on D_n (n=3-13) and E8-related lattices (n=6-9).
• To compare simulation results with theoretical predictions from series expansions.
• To analyze the behavior of percolation thresholds and scaling exponents in high dimensions.

Main Methods:

• Invasion percolation simulations were employed to estimate percolation thresholds.
• Dimensional series expansion based on lattice animal enumeration was used for comparison on D_n lattices.
• High-dimensional systems were simulated using efficient algorithms.

Main Results:

• Bond percolation thresholds on D_n lattices rapidly approach the Bethe lattice limit as dimensionality (n) increases.
• Corrections to the Bethe lattice limit exhibit unexplained trends.
• The finite-size scaling exponent for invasion percolation is specific to the lattice type and percolation problem (site vs. bond).

Conclusions:

• High-dimensional lattices exhibit distinct percolation behaviors compared to traditional cubic lattices.
• The observed trends in corrections and scaling exponents warrant further theoretical investigation.
• Percolation universality may be more nuanced in high-dimensional, high-connectivity systems.