Percolation dangling ends

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... 

In the above node-link-blob picture, the percolation cluster is self-similar up to a length scale in the sense that starting from the length scale the links contain blobs (and the dangling ends) which, in turn, are composed of links and blobs (and the dangling ends) up to the lowest scale (of the lattice). This self-similarity extends up to infinite scale at the percolation threshold (where becomes infinitely large). 

In this problem there is one point to clarify. If one believes that the SAW can only live on the backbone of percolation cluster (otherwise it will be trapped at the dangling ends) [59,35,34,31], one should use in the expressions above the corresponding fractal characteristics df,df) for the backbone. A more sophisticated expression was proposed in [34] ... 

Figure 3 Schematic sketch of a two-dimensional percolation cluster structural elements are single connecting bonds, dead (or dangling) ends, and loops (blobs) on all scales (this is only an illustration and not a real generated cluster)...

As stated above, topologically-hyperbranched macromolecules are trees such structures are devoid of loops. One can then envisage the whole structure as starting from one central monomer, whose effective fimctionahty in the network is /. In the simplest case the monomers attached to it may either have the same functionality (and thus be boimd to other monomers) or be at the ends of dangling bonds. In this case (the fimctionahty of each bead is either 1 or/) the (percolating) network is a subset of the (infinite) Cayley tree clearly the same holds for the dendrimers. More disorder can also be envisaged this is the case when some monomers in the network have fimctionahties different from 1 or/. We will consider both cases of disorder in the following. 

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