Self-similar self-avoiding structures: Models for polymers.

The hierarchy of m,n-self-similar self-avoiding structures on a lattice is introduced as a model for linear and branched polymers. The self-similarity condition permits exact solutions that use a generating-function renormalization procedure with full accounting for volume exclusion. Illustrative results are given for the 1,2-linear-chain model and for the 1,2-branched-structure model on a honeycomb lattice. Exponents for the dependence of polymer size on monomer number are obtained for these models, with that for branched structures being independent of the (nonzero) fraction of branch points. Exponents also are found involving branching probability and branching activity; the fact that those quantities are found to be distinct even in the dilute branching limit is a manifestation of the long-range nature of the excluded volume interaction.