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 [Formula: see text] 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 [Formula: see text] This also establishes the leading large-n behavior [Formula: see text] that may be expected on general grounds, where N n is the number of sites on the nth generation tower and [Formula: see text] is the spectral dimension of the fractal.