Percolation backbone dimension

In fact, if one measures the total number of bonds (sites) on the infinite cluster at the percolation threshold (pc) in a (large) box of linear size L, then this number or the mass of the infinite cluster will be seen to scale with L as where die (< d) is called the fractal dimension of the infinite cluster at the percolation threshold. Similar measurements for the backbone (excluding the dangling ends of the infinite cluster) give the backbone mass scaling as, de < die, where dfi is called the backbone (fractal) dimension. In fact, die can be very easily related to the embedding Euclidean dimension d of the cluster by... 

After having discussed the behavior of SAWs on deterministic fractals, we move on to the second major topic of this chapter, namely the numerical study of SAWs on random fractals, the latter modelled by percolation. As non-trivial changes in the exponents characterizing the structure of SAWs on the incipient percolation cluster (and as a consequence on its backbone) axe only expected at criticality [5], i.e. for probability p of available sites being p = Pc, the following discussion is restricted to this case. A summary of exponents and fractal dimensions characterizing critical percolation is given in Table 1. 

For a random walk in a plane or in three dimensions, = 2 but for a random walk along the lines of another random walk, = 4. On a percolating cluster, at the threshold, lies between 2 and 4, which means that it will be explored a little more quickly than would the structure of a random walk, but a little more slowly than would a plane or a 3-dimensional space. The dead branches of the cluster are like traps the walker must escape from in order to make progress. However, if the walker is located on the backbone of the cluster, much better progress can be made than would be possible through the structure of a random walk because, at the threshold, the backbone of a spanning cluster is more string-like than the latter. 

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