On the free energy density of factor models on biregular graphs

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 $$H_G(x)=\sum _{f\in F}h\left(\sum _{v\in \mathrm{\partial }f}{x}_v\right),$$ 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.