Polymer statistics, then-vector model, and thermodynamic stability

The scaling hypothesis is now widely used in polymer statistics because of the correspondence between this problem and the n→0 limit of the n-vector model of magnetism. We prove here that if the n→0 vector model satisfies scaling, then its free energy does not always satisfy the usual convexity requirements of thermodynamic stability, even above the critical temperature in small fields. We also show that, because of the very different role played by the field variables in the two interpretations, the corresponding model of polymer solutions does not suffer from this instability.