Models for collapse in trees and c-animals

The authors consider the phenomenon of collapse in a lattice model of a branched polymer in a dilute solution. They model the polymer as a lattice animal with a fixed cyclomatic index (c), and consider two alternative but equivalent interpretations in which the collapse is driven either by the contact interaction between non-bonded nearest neighbours, or by the decreasing quality of the solvent. They prove that the reduced limiting free energy is independent of c. For the contact model, this implies that, if a collapse transition exists at beta = beta 0 (where ebeta is the fugacity) for trees (c=0), then a collapse transition occurs at beta 0 for all values of c. The critical point of the solvent model is - beta 0/2, independent of c and, moreover, the critical exponent alpha is the same for both models, independent of c. They use these results to improve their numerical estimates of the temperature dependence of the limiting free energy and confirm their earlier estimate of the value of the cross-over exponent phi .