Recurrence and trapping on a fractal solid

• Published in: Chemical physics letters 1999-06, Vol.306 (5), p.411-415
• Main Authors: Garza-López, Roberto A., Ngo, Minh, Delgado, Erica, Kozak, John J.
• Format: Article
• Language: English
• ISSN: 0009-2614; 1873-4448
• DOI: 10.1016/S0009-2614(99)00479-0

We consider the `second-generation' 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-generation' 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.