Correct correspondence between self-avoiding random walks and a magnetic system asn→0on a lattice

It is pointed out that if one considers the $n=0\mathrm{}$ limit of a magnetic system consisting of $n$-component classical spins on a lattice, one indeed obtains a correspondence with a system of self-avoiding random walks, that is, polymer chains with excluded volume in a solution, but the correspondence is not isomorphic. It turns out that $K$ and $H$ do not serve as the activities for the polymer system, which in fact are given by $\frac{K}{z}$ and $\frac{H}{\sqrt{z}}$, where $z=1+\frac{H^2}{2}$. Moreover, the polymer free energy $\hat{W}$ is also different from the magnetic free energy $W_0$ in the limit $n\to 0$. It is also pointed out that $\hat{W}$ satisfies the proper convexity properties, even though $W_0$ need not.