Ising models on locally tree-like graphs

TL;DR

This paper proves that for ferromagnetic Ising models on graphs that resemble trees locally, the cavity method accurately predicts free energy and local marginals, validated through convergence to infinite random trees.

Contribution

It establishes the correctness of the cavity prediction for free energy and local marginals on locally tree-like graphs at any positive temperature and external field.

Findings

Cavity prediction for free energy is validated.

Local marginals can be approximated via mean field equations.

Results hold for graphs converging to trees, including random regular and bounded degree graphs.

Abstract

We consider ferromagnetic Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that the "cavity" prediction for the limiting free energy per spin is correct for any positive temperature and external field. Further, local marginals can be approximated by iterating a set of mean field (cavity) equations. Both results are achieved by proving the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on the appropriate infinite random tree.