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.