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Bounding the partition function of spin-systems

With a graph $G=(V,E)$ we associate a collection of non-negative real weights $\cup_{v\in V}{λ_{i,v}:1\leq i \leq m} \cup \cup_{uv \in E} {λ_{ij,uv}:1\leq i \leq j \leq m}$. We consider the probability distribution on ${f:V\rightarrow{1,...,m}}$ in which each $f$ occurs with probability proportional to $\prod_{v \in V}λ_{f(v),v}\prod_{uv \in E}λ_{f(u)f(v),uv}$. Many well-known statistical physics models, including the Ising model with an external field and the hard-core model with non-uniform activities, can be framed as such a distribution. We obtain an upper bound, independent of $G$, for the partition function (the normalizing constant which turns the assignment of weights on $\{f:V\rightarrow{1,...,m\}}$ into a probability distribution) in the case when $G$ is a regular bipartite graph. This generalizes a bound obtained by Galvin and Tetali who considered the simpler weight collection $\{λ_i:1 \leq i \leq m\} \cup \{λ_{ij}:1 \leq i \leq j \leq m\}$ with each $λ_{ij}$ either 0 or 1 and with each $f$ chosen with probability proportional to $\prod_{v \in V}λ_{f(v)}\prod_{uv \in E}λ_{f(u)f(v)}$. Our main tools are a generalization to list homomorphisms of a result of Galvin and Tetali on graph homomorphisms and a straightforward second-moment computation.