n→∞Limit ofO(n)Ferromagnetic Models on Graphs

Abstract

Thirty years ago, H. E. Stanley showed that an $O\left(n\right)$ 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 This fundamental result is no longer valid on more general discrete structures lacking in translation invariance, i.e., on graphs.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 for ferromagnetic models such singular parts still coincide even on graphs in the thermodynamic limit. This implies that the critical exponents of $O\left(n\right)$ models on graphs for $n\to \infty$ tend to the spherical ones and depend only on the graph spectral dimension.