Caley tree

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

We have already mentioned that the lattice structure, while used for most percolation studies, is not really necessary and that even without the help of a lattice the critical exponents seem to have invariable lattice values. According to the simple classical theory this is not the case since die radius of trees on a periodic lattice (with excluded volume effects) increase for large cluster masses s at least with s (in d dimensions) whereas in the classical theory on a continuum a Caley tree has a radius varying asymptotically with s, independent of d. 

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