Recurrence and trapping on a fractal solid

We consider the 'second-generation' Menger sponge, a symmetric fractal of d imension d approximate to 2.7268..., and associated lattice of N = 1056 sit es and uniform coordination number v = 4, and calculate the mean walklength [n] before trapping for a random walker on this lattice. In addition to st rengthening results obtained previously, in which we examined whether value s of (n) calculated for various configurations of the 'first-generation' Me nger sponge (N = 72) are intermediate between those calculated for the corr esponding d = 2- and 3-dimensional lattices, we demonstrate here that a cla ssic result of Montroll on recurrence times is more general than had previo usly been reported. In particular, we show for the lattice studied here tha t the expected walklength (n) conditional on starting from a site nearest-n eighbor to the point of origin is given by (N - 1) exactly. (C) 1999 Elsevi er Science B.V. All rights reserved.