A $q$-state spin system is defined by an underlying $n$-vertex graph~$G$ together with one or more parameters. Configurations of a spin system are assignments $\sigma:V(G)\to[q]$ of spins to the vertices (or sometimes edges) of~$G$. Each configuration has a defined weight, which, when normalised by the partition function of the system, specifies the Gibbs distribution on configurations. Generally, as some parameter (sometimes called `temperature') is varied, the system undergoes a phase transition. On one side, the (complex) zeros of the partition function avoid the real axis and Glauber (single site) dynamics mixes rapidly; on the other, zeros approach the real axis and the mixing time of Glauber dynamics is exponential in the size of the underlying graph.

A rare exception to this picture is the monomer-dimer model, which does not exhibit a phase transition at any non-zero temperature (Heilmann and Lieb). However, it is possible to obtain interesting examples from other models --- antiferromagnetic Ising model, hard-core model --- by suitably restricting the underlying graph~$G$. I'll explore this phenomenon mainly in the context of the hard-core model, as it has the most intricate and fascinating behaviour.