On the free energy density of factor models on biregular graphs

TL;DR

This paper proves that for large girth sequences of biregular factor graphs, the free energy density converges and is accurately approximated by the Bethe-approximation, extending understanding of statistical mechanics on such graphs.

Contribution

It establishes the convergence of free energy density for large girth biregular factor graphs and confirms the Bethe-approximation as the limit.

Findings

Free energy density converges for large girth biregular graphs.

Bethe-approximation accurately predicts the limiting free energy.

Results apply to symmetric concave Hamiltonians.

Abstract

Let $h(0),h(1),\dots,h(k)$ be a symmetric concave sequence. For a $(d,k)$-biregular factor graph $G$ and $x\in \{0,1\}^V$, we define the Hamiltonian \[H_G(x)=\sum_{f\in F} h\left(\sum_{v\in \partial f} x_v\right),\] where $V$ is the set of variable nodes, $F$ is the set of factor nodes. We prove that if $(G_n)$ is a large girth sequence of $(d,k)$-biregular factor graphs, then the free energy density of $G_n$ converges. The limiting free energy density is given by the Bethe-approximation.