  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.
