The standard hard-core model on a locally finite graph describes a family of Gibbs specifications, parameterized by the so-called activity parameter. An important problem is to determine when the model has a unique Gibbs measure and when it exhibits a phase transition (that is, has multiple Gibbs measures). Such models arise in combinatorics, statistical physics and communication networks, and recent work has uncovered interesting connections between phase transitions in these models and the hardness of various counting problems. We will describe phase transition properties of the hard-core model and its various avatars, including a continuous version of the model and describe some interesting consequences.
