Fractal to Euclidean crossover and scaling for random walks on percolation

We perform random walk simulations on binary three-dimensional simple cubic lattices covering the entire ratio of open/closed sites (fractionp) from the critical percolation threshold to the perfect crystal. We observe fractal behavior at the critical point and derive the value of the number-of-sites-visited exponent, in excellent agreement with previous work or conjectures, but with a new and imprOVed computational algorithm that extends the calculation to the long time limit. We show the crossover to the classical Euclidean behavior in these lattices and discuss its onset as a function of the fractionp. We compare the observed trends with the two-dimensional case.