  I review recent results and unsolved problems concerning the hard-core
lattice gas and the q-coloring model (antiferromagnetic Potts model at zero
temperature). For each model, I consider its equilibrium properties
(uniqueness/nonuniqueness of the infinite-volume Gibbs measure, complex zeros
of the partition function) and the dynamics of local and nonlocal Monte Carlo
algorithms (ergodicity, rapid mixing, mixing at complex fugacity). These
problems touch on mathematical physics, probability, combinatorics and
theoretical computer science.
