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 ${T}^{(n)}=[{3}^{n}{5}^{n+1}{+4(5}^{n})\ensuremath{-}{3}^{n}]{/(3}^{n+1}+1)$ 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 $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}$ of the gasket, the precise asymptotic behavior for large ${N}_{n}$ is ${T}^{(n)}\ensuremath{\rightarrow}{1/3(2N}_{n}{)}^{2/\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}\ensuremath{\sim}{N}^{1.464}.$ This serves as a partial check on our result, as it is (a) intermediate between the known results $T\ensuremath{\sim}{N}^{2}$ $(d=1)$ and $T\ensuremath{\sim}N\mathrm{ln}N$ $(d=2)$ 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.