Structural and dynamical properties of the percolation backbone in two and three dimensions

We study structural and dynamical properties of the backbone of the incipient infinite cluster for site percolation in two and three dimensions. We calculate the average mass of the backbone in chemical l space, 〈MB(l)〉~l^dₗB, where dₗB is the chemical dimension. We find dₗB=1.45±0.01 in $d=2$ and dₗB=1.36±0.02 in $d=3$. The fractal dimension in r space dfB is obtained from the relation dfB=dₗBdₘᵢₙ, dfB=1.64±0.02 in $d=2$ and dfB=1.87 ± 0.03 in $d=3$, where dₘᵢₙ is the fractal dimension of the shortest path. The distribution function ΦB(r,l) is determined, giving the probability of finding two backbone sites at the spatial distance r connected by the shortest path of length l, as well as the related quantity lₘᵢₙB(r,Nₐᵥ), giving the length of the minimal shortest path for two backbone sites at distance r as a function of the number Nₐᵥ of configurations considered. Regarding dynamical properties, we study the distribution functions PB(l,t) and PB(r,t) of random walks on the backbone, giving the probability of finding a random walker after t time steps, at a chemical distance l, and Euclidean distance r from its starting point, respectively, and their first moments 〈lB(t)〉~t^1/dwBl and 〈rB(t)〉~t^1/dwB, from which the fractal dimensions of the random walk dwBl and dwB are estimated. We find dwBl=2.28±0.03 and dwB=2.62±0.03 in $d=2$ as well as dwBl=2.25±0.03 and dwB=3.09±0.03 in $d=3$.