Renormalization of first-passage times for random walks on deterministic fractals

The first-passage time density for nearest-neighbor random walks on various deterministic fractal lattices is calculated. The procedure consists of the derivation of a renormalization equation, similar to that derived by Machta for a random walk in one dimension [Phys. Rev. B 24, 5260 (1981)], and in its solution in terms of an auxiliary function, defined by a functional equation. Both the cases of waiting times with finite first moment or with fractal time properties are discussed. By rescaling time and length in the appropriate way, the first-passage time density approaches a ``universal'' asymptotic form {ψ}({τ}). The scaling properties are consistent with subdiffusive behavior 〈{δ}r²(t)〉{~}t2/dw, with dw>2, as discussed previously in the literature. The overall shape of {ψ}({τ}) is found to be very similar to that for a random walk in one dimension, except that the small-{τ} behavior is qualitatively different in both cases. The existence of these two time regimes was already suggested by Guyer [Phys. Rev. A 29, 2751 (1984)] and explains why different scaling forms for the Green's function have been proposed in the literature.