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

TL;DR

This paper extends the classical result of the O(n) model's convergence to the spherical model from lattices to general graphs, showing that critical behavior and exponents depend only on the graph's spectral dimension.

Contribution

It demonstrates that the singular parts of free energies of O(n) models on graphs coincide with spherical models as n approaches infinity, generalizing prior lattice results.

Findings

Singular parts of free energies coincide on graphs in the thermodynamic limit.

Critical exponents of O(n) models on graphs tend to spherical model exponents.

Critical behavior depends only on the spectral dimension of the graph.

Abstract

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 This fundamental result, providing the basis for more detailed studies of continuous symmetry spin models, is no longer valid on more general discrete structures lacking of translation invariance, i.e. on graphs.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.