Magnetic analog of self-avoiding random walks (polymer chains) on a lattice

It is shown that if one considers the n→0 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 K/z and H√z, where z=1+H²2. Moreover, the polymer free energy ^W is also different from the magnetic free energy W₀ in the limit n→0. Various polymer correlation functions are calculated and are found to be different from those proposed by other authors. It is also shown that ^W satisfies the proper convexity properties, even though W₀ does not.