Size and overlap of randomly branched polymers

The correlation length for percolation (and gelation) is the size of the characteristic branched polymer with N monomers. Randomly branched polymers are fractals, so the size R and the number of monomers A in a 

The same relation is valid for the characteristic branched polymer  

Since N diverges at the gel point with exponent /a [see Eq. (6.95)], the correlation length must also diverge  

The exponentt v describing the divergence of the correlation length is related to a and the fractal dimension V by combining Eqs (6.95) and 

In mean field, (7=1/2 and T = 4 (see Section 6.4) so i/ = 1 /2. For three-dimensional percolation, this critical exponent is iy — 0.88 and the fractal dimension of randomly branched polymers in the polymerization reactor is P= 2.53. 

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