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, $d_{\mathrm{u}}=6$. Using theory and simulations, we consider the scaling regime and obtain that both caging and subdiffusion scale logarithmically for $d\ge d_{\mathrm{u}}$. The theoretical derivation, which considers Bethe lattices with generalized connectivity and a random graph model, confirms that logarithmic scalings should persist in the limit $d\to \infty$. The computational validation employs accelerated random walk simulations with a transfer-matrix description of diffusion to evaluate directly the dynamical critical exponents below $d_{\mathrm{u}}$ as well as their logarithmic scaling above $d_{\mathrm{u}}$. Our numerical results improve various earlier estimates and are fully consistent with our theoretical predictions.