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

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.

arXiv comment12 pages, 6 figures

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Cited by in corpus (3)

• Interplay between percolation and glassiness in the random Lorentz gas
• High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas
• Percolation thresholds on high dimensional $D_n$ and dense packing lattices