Exact Formula for the Mean Length of a Random Walk on the Sierpinski Tower

We consider an unbiased random walk on a finite, nth generation Sierpinski gasket (or "tower") in d = 3 Euclidean dimensions, in the presence of a trap at one vertex. The mean walk length (or mean number of time steps to absorption) is given by the exact formula The generalization of this formula to the case of a tower embedded in an arbitrary number d of Euclidean dimensions is also found, and is given by This also establishes the leading large-n behavior that may be expected on general grounds, where Nn is the number of sites on the nth generation tower and is the spectral dimension of the fractal.