The $n\to\infty$ limit of O(n) model on graphs

Thirty years ago, Stanley showed that an O(n) spin model on a lattice tends to a spherical model as $n\to\infty$. This means that at any temperature the corresponding free energies coincide. This fundamental result, providing the basis for more detailed studies of continuous symmetry spin models, is no longer valid on more general discrete structures lacking of translation invariance, i.e. on graphs. However only the singular parts of the free energies determine the critical behavior of the two statistical models. Here we show that such singular parts still coincide even on general graphs in the thermodynamic limit. This implies that the critical exponents of O(n) models on graphs for $n\to\infty$ tend to the spherical ones and therefore they only depend on the graph spectral dimension.