Analytic expression for the mean time to absorption for a random walker on the Sierpinski gasket

The exact analytic expression for the mean time to absorption (or mean walk length) for a particle performing a random walk on a finite Sierpinski gasket with a trap at one vertex is found to be T-(n) =[3(n)5(n+1) + 4(5(n)) - 3(n)]/(3(n+1) + 1) where n denotes the generation index of the gasket, and the mean is over a set of starting points of the walk distributed uniformly over all the other sites of the gasket. In terms of the number N-n of sites on the gasket and the spectral dimension (d) over tilde of the gasket, the precise asymptotic behavior for large N-n is T-(n)-->1/3(2N(n))(2/(d) over tilde)similar toN(1.464). This serves as a partial check on our result, as it is (a) intermediate between the-known results Tsimilar toN(2) (d= 1) and Tsimilar toN In N (d= 2) for random walks on d-dimensional Euclidean lattices and (b) consistent with the known result for the asymptotic behavior of the mean number of distinct sites visited in a random walk on a fractal lattice.