Dynamics around the site percolation threshold on high-dimensional hypercubic lattices.

Area of Science:

• Physics
• Statistical Mechanics
• Complex Systems

Background:

• Classical percolation models are foundational to understanding disordered systems.
• Recent advances in the glass problem necessitate a reevaluation of these models.
• The behavior of systems near the percolation threshold is of significant interest.

Purpose of the Study:

• To investigate ant displacement and subdiffusion in cubic lattices near the percolation threshold.
• To explore scaling regimes both below and above the upper critical dimension (d_u=6).
• To theoretically and computationally validate logarithmic scaling phenomena.

Main Methods:

• Theoretical derivation using Bethe lattices with generalized connectivity and random graph models.
• Computational validation via accelerated random walk simulations.
• Transfer-matrix description of diffusion to evaluate dynamical critical exponents.

Main Results:

• Observed logarithmic scaling for both caging and subdiffusion for d≥d_u.
• Theoretical confirmation that logarithmic scalings persist in the limit d→∞.
• Numerical results improve upon previous estimates and align with theoretical predictions.

Conclusions:

• Logarithmic scaling is a key characteristic of percolation systems above the upper critical dimension.
• The study provides robust theoretical and numerical evidence for these scaling behaviors.
• Findings contribute to a deeper understanding of disordered systems and the glass problem.