Dynamics around the Site Percolation Threshold on High-Dimensional Hypercubic Lattices

TL;DR

This paper investigates the dynamics of a random walk near the percolation threshold on high-dimensional cubic lattices, revealing logarithmic scaling behaviors above the upper critical dimension through theory and simulations.

Contribution

It provides a combined theoretical and computational analysis of percolation dynamics, extending understanding of scaling behaviors above the upper critical dimension.

Findings

Logarithmic scaling of caging and subdiffusion for d >= 6

Theoretical derivation using Bethe lattices and random graph models

Numerical validation with accelerated random walk simulations

Abstract

Recent advances on the glass problem motivate reexamining classical models of percolation. Here, we consider the displacement of an ant in a labyrinth near the percolation threshold on cubic lattices both below and above the upper critical dimension of simple percolation, d_u=6. Using theory and simulations, we consider the scaling regime part, and obtain that both caging and subdiffusion scale logarithmically for d >= d_u. The theoretical derivation considers Bethe lattices with generalized connectivity and a random graph model, and employs a scaling analysis to confirm that logarithmic scalings should persist in the infinite dimension limit. The computational validation employs accelerated random walk simulations with a transfer-matrix description of diffusion to evaluate directly the dynamical critical exponents below d_u as well as their logarithmic scaling above d_u. Our numerical results improve various earlier estimates and are fully consistent with our theoretical predictions.