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 T-(n) = 9[4(n)6(n+1) + 5(6)(n) - 4(n)]/5(4(n+1) + 2). 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 T-(n) = d(2)[(d + 1)(n) (d + 3)(n+1) + (d + 2)(d + 3)(n) - (d + 1)(n)]/ (d + 2)[d + 1)(n+1) + d - 1]. This also establishes the leading large-n behavior T-(n) similar to N-n(2/(d) over bar) that may be expected on general grounds, where N-n is the number of sites on the nth generation tower and (d) over tilde = ln(d + 1)(2)/ln(d + 3) is the spectral dimension of the fractal.