A new mapping between self-avoiding walks and the n to 0 limit

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Abstract
It is shown that by considering a new mapping between self-avoiding walks and the n to 0 limit of the n-component classical spin system with the constraint that each spin has a fixed length square root n(S2=n), one can study a grand canonical ensemble of self-avoiding walks of all lengths, including those with zero lengths. It is shown that this new mapping is also not isomorphic. Moreover, the mapping is physically meaningful only for H<or= square root 2: higher values of H in the magnetic system do not produce a meaningful analogy with self-avoiding walks. The correlation functions for self-avoiding walks can be shown to be different from those proposed by other authors.