The replica symmetric solution for Potts models on d-regular graphs

TL;DR

This paper derives an explicit formula for the free energy of ferromagnetic Potts models on sparse, d-regular graphs, confirming the replica symmetric solution and its bounds across all temperature regimes.

Contribution

It provides a rigorous, explicit formula for the free energy of Potts models on d-regular graphs, validating the replica symmetric Bethe approximation.

Findings

The formula matches the Bethe free energy at fixed points of belief propagation.

The replica symmetric Bethe formula bounds the free energy for all models with permissive interactions.

The results hold for all temperature regimes on graphs converging locally to the d-regular tree.

Abstract

We provide an explicit formula for the limiting free energy density (log-partition function divided by the number of vertices) for ferromagnetic Potts models on uniformly sparse graph sequences converging locally to the d-regular tree for d even, covering all temperature regimes. This formula coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation recursion on the d-regular tree, the so-called replica symmetric solution. For uniformly random d-regular graphs we further show that the replica symmetric Bethe formula is an upper bound for the asymptotic free energy for any model with permissive interactions.