Recurrence and trapping on a fractal solid

We consider the `second-generationʹ Menger sponge, a symmetric fractal of dimension d≈2.7268…, and associated lattice of N=1056 sites and uniform coordination number v=4, and calculate the mean walklength 〈n〉 before trapping for a random walker on this lattice. In addition to strengthening results obtained previously, in which we examined whether values of 〈n〉 calculated for various configurations of the `first-generationʹ Menger sponge (N=72) are intermediate between those calculated for the corresponding d=2- and 3-dimensional lattices, we demonstrate here that a classic result of Montroll on recurrence times is more general than had previously been reported. In particular, we show for the lattice studied here that the expected walklength 〈n〉 conditional on starting from a site nearest-neighbor to the point of origin is given by (N−1) exactly.