POLYMER MODELS AND GENERALIZED POTTS-KASTELEYN MODELS

The well-established relation between Potts models with nu spin values and random-cluster models (with intracluster bonding favored over int ercluster bonding by a factor nu) is explored, but with the random-clu ster model replaced by a much generalized polymer model, implying a co rresponding generalization of the Potts model. The analysis is carried out in terms a given defined function R(rho), an entropy/free-clergy density for the polymer model in the case nu = 1, expressed as a funct ion of the density rho of units. The aim of the analysis is to determi ne the analog R(nu)(rho) of R(rho) for general nonnegative nu in terms of R(rho), and thence to determine the critical value of density rho( nug) at which gelation occurs. This critical value is independent of n u up to a value nu(P), the Potts-critical value. What is principally r equired of R(rho) is that it should show a certain given concave/conve x behavior, although differentiability and another regularizing condit ion are required for complete conclusions. Under these conditions the unique evaluation of R(nu)(rho) in terms of R(rho) is given in a form known to hold for integral nu but not previously extended. The analysi s is carried out in terms of the Legendre transforms of these function s, in terms of which the phenomena of criticality (gelation) and Potts criticality appear very transparently and nu(P) is easily determined. The value of nu(P) is 2 under mild conditions on R. Special interest attaches to the function R0(rho), which is shown to be the greatest co ncave minorant of R(rho). The naturalness of the approach is demonstra ted by explicit treatment of the first-shell model.