High density phases of hard-core lattice particle systems

In this talk, I will discuss the behavior of hard-core lattice particle systems at high fugacities. I will first present a collection of models in which the high fugacity phase can be understood by expanding in powers of the inverse of the fugacity. I will then discuss a model in which this expansion diverges, but which can still be solved by expanding in other high fugacity variables. This model is an interacting dimer model, introduced by O.Heilmann and E.H.Lieb in 1979 as an example of a nematic liquid crystal. Heilmann and Lieb proved that the dimers spontaneously align, and conjectured that they do not exhibit long range positional order. I will present a recent proof of this conjecture.