An exact relation between the classical n-vector model ferromagnet and the self-avoiding walk problem

Diagrams are used to present an exact high-temperature low-field series expansion for the Gibbs free energy of the classical n-vector model ferromagnet. It is shown, within the framework of the series expansion method, that, to leading order in (small) n, the zero-field Gibbs free energy and the zero-field susceptibility of the ferromagnet are effectively equal to the self-avoiding polygon and self-avoiding chain generating functions, respectively. The critical point and critical exponents determined by the above properties of the ferromagnet to leading order in n are identified with those of the corresponding self-avoiding walk problem.