Bounding the partition function of spin-systems

With a graph $G=(V,E)$ we associate a collection of non-negative real weights $\cup_{v\in V}{\lambda_{i,v}:1\leq i \leq m} \cup \cup_{uv \in E} {\lambda_{ij,uv}:1\leq i \leq j \leq m}$. We consider the probability distribution on $f:V\to 1,\mathrm{.}\mathrm{.}\mathrm{.},m$ in which each $f$ occurs with probability proportional to $\prod_{v \in V}\lambda_{f(v),v}\prod_{uv \in E}\lambda_{f(u)f(v),uv}$. 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 $\{f:V\to 1,\mathrm{.}\mathrm{.}\mathrm{.},m\}$ 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 ${\lambda_i:1 \leq i \leq m} \cup {\lambda_{ij}:1 \leq i \leq j \leq m}$witheach$\lambda_{ij}$either0or1andwitheach$f$chosenwithprobabilityproportionalto$\prod_{v \in V}\lambda_{f(v)}\prod_{uv \in E}\lambda_{f(u)f(v)}$. 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.