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.Comment: 12 pages, 6 figure
