Bounding the partition function of spin-systems

Abstract

With a graph $G=(V,E)$ we associate a collection of non-negative real weights ${\cup }_{v\in V}{\lambda }_{i,v}:1\le i\le m\cup {\cup }_{uv\in E}{\lambda }_{ij,uv}:1\le i\le j\le m$. We consider the probability distribution on $f:V\to 1,...,m$ 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 $\{f:V\to 1,...,m\}$ 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 $\{{\lambda }_i:1\le i\le m\}\cup \{{\lambda }_{ij}:1\le i\le j\le m\}$ 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.

13 pages. Appeared in Electronic Journal of Combinatorics in 2006