Polymers and random graphs: Asymptotic equivalence to branching processes

In 1974, Falk and Thomas did a computer simulation of Flory's Equireactive RA_f Polymer model, rings forbidden and rings allowed. Asymptotically, the Rings Forbidden model tended to Stockmayer's RA_f distribution (in which the sol distribution sticks after gelation), while the Rings Allowed model tended to the Flory version of the RA_f distribution. In 1965, Whittle introduced the Tree and Pseudomultigraph models. We show that these random graphs generalize the Falk and Thomas models by incorporating first-shell substitution effects. Moreover, asymptotically the Tree model displays postgelation sticking. Hence this phenomenon results from the absence of rings and occurs independently of equireactivity. We also show that the Pseudomultigraph model is asymptotically identical to the Branching Process model introduced by Gordon in 1962. This provides a possible basis for the Branching Process model in standard statistical mechanics.