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