Spherical limit of $n$-vector correlations

It is shown that for any summable translationally invariant interaction the correlation functions of any order of the classical Heisenberg model ($n$-vector model) as $n\to\infty$ and for any fixed constant temperature $T$ converge to the corresponding correlation functions of the Berlin–Kac spherical model. A simple proof of the equality of the free energies of these models in the limit $n\to\infty$ is obtained in the process. The form that the result will take in the case without translational invariance is indicated.