Random walks on the Menger sponge

We show that values of the mean walklength 〈n〉 before trapping for a random walker on a Menger sponge of N sites, coordination number ν = 4 and fractal dimension d ∼ 2.7268, are intermediate between results calculated for walks on the corresponding N = l × m torus (ν = 4 and d = 2) and the d = 3 simple cubic lattice.