Scaling of the first-passage time and the survival probability on exact and quasi-exact self-similar structures

The authors calculate the asymptotic behaviour of the moments of the first-passage time and survival probability for random walks on an exactly self-similar tree, and on a quasi-self-similar comb, by applying an exact decimation approach to the master equations. For the hierarchical comb, a transition from ordinary to anomalous diffusion occurs at R=2, where R is the ratio of teeth length in successive iterations of the structure. In the anomalous regime (R>2), the positive integer moments of the first-passage time, (t q ), scale as L tau q , with tau q =1+(2q-1)ln R/ln 2, where L is the linear distance from input to output. The asymptotic behaviour of the survival probability is studied using both scaling theory and by a direct solution of the master equations. They find that the characteristic time, t*, in the asymptotic exponential decay of the survival probability, exp(-t/t*), scales as t* approximately L tau *= , with tau *=ln R 2 /ln 2, i.e. tau * is distinct from tau 1 . However, substantial corrections to this asymptotic form for tau * exist, and these are needed to account for the recent simulation data of Havlin and Matan (1988).