Classical,n-Component Spin Systems or Fields with Negative Even Integraln

An $n$-component classical spin system with arbitrary pair interactions, or a lattice field theory, is considered, in arbitrary dimensionality $d$, with spin weighting factor $\exp [-w(s^2\left)\right]$, or equivalently local interactions satisfying $w\left(y\right)=\mathrm{O}\left(y^{m^{+\delta }}\right)$ as $y\to 0$ with $m=1,2,\dots \mathrm{and} \delta >0\mathrm{}$. Analytic continuation in $n$ is defined and used to show that for $n=-2k$ with $k=1,2,\dots ,m$ the zero-field free energy and all isotropic correlation functions and cumulants take their Gaussian (or free field) values.