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)]similar to l(dB)l, where d(l)(B) is the chemical dimension. We find d(l)(B)=1.45+/-0.01 in d=2 and d(l)(B)=1.36+/-0.02 in d=3. The fractal dimension in r space d(f)(B) is obtained from the relation d(f)(B)=d(l)(B)d(min), d(f)(B)=1.64+/-0.02 in d=2 and d(f)(B)=1.87 +/- 0.03 in d=3, where d(min), is the fractal dimension of the shortest path. The distribution function Phi(B)(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 l(min)(B)(r,N-av), giving the length of the minimal shortest path for two backbone sites at distance r as a function of the number N-av of configurations considered. Regarding dynamical properties, we study the distribution functions P-B(l,t) and P-B(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 [l(B)(t)]similar to t(1/dwBl) and [r(B)(t)]similar to t(1/dwB), from which the fractal dimensions of the random walk d(w)(Bl) and d(w)(B) are estimated. We find d(w)(Bl)=2.28+/-0.03 and d(w)(B)=2.62+/-0.03 in d=2 as well as d(w)(Bl)=2.25+/-0.03 and d(w)(B)=3.09+/-0.03 in d=3.