Polymer models and generalized Potts-Kasteleyn models

The well-established relation between Potts models withv spin values and random-cluster models (with intracluster bonding favored over intercluster bonding by a factorv) is explored, but with the random-cluster model replaced by a much generalized polymer model, implying a corresponding generalization of the Potts model. The analysis is carried out in terms a given defined functionR(ρ), an entropy/free-energy density for the polymer model in the casev=1, expressed as a function of the density ρ of units. The aim of the analysis is to determine the analogRv(ρ) ofR(ρ) for general nonnegativev in terms ofR(ρ), and thence to determine the critical value of density ρvg at which gelation occurs. This critical value is independent ofv up to a valuevP, the Potts-critical value. What is principally required ofR(ρ) is that it should show a certain given concave/convex behavior, although differentiability and another regularizing condition are required for complete conclusions. Under these conditions the unique evaluation ofRv(ρ) in terms ofR(ρ) is given in a form known to hold for integralv but not previously extended. The analysis is carried out in terms of the Legendre transforms of these functions, in terms of which the phenomena of criticality (gelation) and Potts criticality appear very transparently andvP is easily determined. The value ofvP is 2 under mild conditions onR. Special interest attaches to the functionR0(ρ), which is shown to be the greatest concave minorant ofR(ρ). The naturalness of the approach is demonstrated by explicit treatment of the first-shell model.