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, du=6. Using theory and simulations, we consider the scaling regime and obtain that both caging and subdiffusion scale logarithmically for d≥du. 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→∞. The computational validation employs accelerated random walk simulations with a transfer-matrix description of diffusion to evaluate directly the dynamical critical exponents below du as well as their logarithmic scaling above du. Our numerical results improve various earlier estimates and are fully consistent with our theoretical predictions.