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

Abstract

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 of the gasket, the precise asymptotic behavior for large N(n) is T((n))-->1/3(2N(n))(2/d)-N1.464. This serves as a partial check on our result, as it is (a) intermediate between the known results T-N2 (d=1) and T-N ln 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.