Exact enumeration approach to fractal properties of the percolation backbone and 1/σ expansion

An exact enumeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold. We use this approach to calculate exactly the first low-density expansion of LBB@) for arbitrary system dimensionality d, where L B B @ ) is the mean number of backbone bonds and p is the bond occupation probability. Standard series extrapolation methods provide estimates of the fractal dimension of the backbone for all d ; these disagree with the Sierpinski gasket model of the backbone. We also calculate the first low-density expansions of Lmin@) and L e d @ ) which are, respectively, the mean number of bonds in the minimum path between i and i and the mean number of singly connected (‘red’) bonds. How can one describe the flow of fluid in random porous media? This important question has long eluded explanation. Recently, considerable attention has been focused on the utility of fractals as models of random media. In particular, the topology of the network that exists just at the onset of fluid flow has been modelled by percolation theory. Bonds are considered intact if fluid can flow through them. When the fraction of bonds is small, the system consists of many small finite clusters. However, as the bond fraction approaches a critical value pc the clusters grow large and ramified until at pc fluid can flow. If we consider the network of intact bonds right at pc, there will be a subset of bonds that carry fluid (‘backbone’ bonds) and a remainder that does not (‘dangling ends’). The structure of the backbone remains an important open question. Two models of the backbone have been discussed in the literature. In one (Gefen et a1 1981), the backbone is replaced by a d-dimensional Sierpinski gasket. In the other (Stanley 1977, Coniglio 1981, 1982, Pike and Stanley 1981, Stanley and Coniglio 1983), the backbone is considered to consist of an alternating sequence of singly connected (‘red’) bonds and multiply connected (‘blue’) bonds; these are shown in colour as figure 5 of Hamann (1983). The advantage of the Sierpinski gasket model of the backbone is that one can calculate exactly its fractal dimension, DB = ln(d + l>/ln 2. Hence it is important to obtain estimates of DB for the actual backbone of percolation clusters. Thus far, the only efforts have been Monte Carlo simulations in d = 2, 3 for the backbone order parameter exponent Pe; however, the order parameter is extremely difficult to calculate by Monte Carlo methods (Kirkpatrick 1978, Li and Streider 1982). Also, a limited attempt has been made to estimate the field-like scaling power yl, by large-cell t Supported in part by grants from NSF, ARO and ONR. @ 1983 The Institute of Physics L475 L476 Letter to the Editor position-space renormalisation group; however, this work was limited to d = 2. The Sierpinski gasket model gives reasonable quantitative values for d = 2, 3 but not for higher d. For this and other reasons, it is highly desirable to have calculations of df for the backbone for general d, in order that one can better assess the relative merits of various models of backbone topology. To this end, here we present an exact enumeration approach for the backbone fractal, and calculate the first ten terms in the low-density expansions for arbitrary d . We shall see that these ten terms behave sufficiently smoothly with increasing order that extrapolations to obtain the asymptotic behaviour can be made (table 1) . Table 1. Critical exponents characterising the backbone of the incipient infinite cluster in percolation. The basic quantities calculated are ls and lmin, since ired = 1 for all d (Coniglio 1982). In order to obtain the exponent for the derived quantities LBB(p) and Lmin@), we need yp, which is also tabulated. Finally, to obtain the backbone fractal dimension, De, we need Y. YP U 5 B 5min P B In(d + 1) D ” = y DB =In2