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{\bf David J. Galvin}
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{\bf Bounding the Partition Function of Spin-Systems}
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With a graph $G=(V,E)$ we associate a collection of non-negative real
weights $\bigcup_{v\in V}\{\lambda_{i,v}:1\leq i \leq m\} \cup
\bigcup_{uv \in E} \{\lambda_{ij,uv}:1\leq i \leq j \leq m\}.$ We
consider the probability distribution on
$\{f:V\rightarrow\{1,\ldots,m\}\}$ 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 $\{f:V\rightarrow\{1,\ldots,m\}\}$ 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 $\{\lambda_i:1 \leq i \leq
m\} \cup \{\lambda_{ij}:1 \leq i \leq j \leq m\}$ with each
$\lambda_{ij}$ either $0$ or $1$ and with each $f$ chosen with
probability proportional to $\prod_{v \in V}\lambda_{f(v)}\prod_{uv
\in E}\lambda_{f(u)f(v)}$. 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.
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